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## G = C7×C23.36C23order 448 = 26·7

### Direct product of C7 and C23.36C23

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C7×C23.36C23
 Chief series C1 — C2 — C22 — C2×C14 — C22×C14 — C7×C22⋊C4 — C7×C22.D4 — C7×C23.36C23
 Lower central C1 — C22 — C7×C23.36C23
 Upper central C1 — C2×C28 — C7×C23.36C23

Generators and relations for C7×C23.36C23
G = < a,b,c,d,e,f,g | a7=b2=c2=d2=e2=1, f2=d, g2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ebe=bd=db, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, eg=ge, fg=gf >

Subgroups: 322 in 234 conjugacy classes, 154 normal (62 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C14, C14, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C28, C28, C2×C14, C2×C14, C2×C14, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C422C2, C2×C28, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C22×C14, C23.36C23, C4×C28, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C22×C28, C22×C28, D4×C14, D4×C14, Q8×C14, C2×C4×C28, C7×C42⋊C2, D4×C28, D4×C28, Q8×C28, C7×C4⋊D4, C7×C22⋊Q8, C7×C22.D4, C7×C4.4D4, C7×C42.C2, C7×C422C2, C7×C23.36C23
Quotients: C1, C2, C22, C7, C23, C14, C4○D4, C24, C2×C14, C2×C4○D4, C22×C14, C23.36C23, C7×C4○D4, C23×C14, C14×C4○D4, C7×C23.36C23

Smallest permutation representation of C7×C23.36C23
On 224 points
Generators in S224
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)(22 23 24 25 26 27 28)(29 30 31 32 33 34 35)(36 37 38 39 40 41 42)(43 44 45 46 47 48 49)(50 51 52 53 54 55 56)(57 58 59 60 61 62 63)(64 65 66 67 68 69 70)(71 72 73 74 75 76 77)(78 79 80 81 82 83 84)(85 86 87 88 89 90 91)(92 93 94 95 96 97 98)(99 100 101 102 103 104 105)(106 107 108 109 110 111 112)(113 114 115 116 117 118 119)(120 121 122 123 124 125 126)(127 128 129 130 131 132 133)(134 135 136 137 138 139 140)(141 142 143 144 145 146 147)(148 149 150 151 152 153 154)(155 156 157 158 159 160 161)(162 163 164 165 166 167 168)(169 170 171 172 173 174 175)(176 177 178 179 180 181 182)(183 184 185 186 187 188 189)(190 191 192 193 194 195 196)(197 198 199 200 201 202 203)(204 205 206 207 208 209 210)(211 212 213 214 215 216 217)(218 219 220 221 222 223 224)
(1 38)(2 39)(3 40)(4 41)(5 42)(6 36)(7 37)(8 221)(9 222)(10 223)(11 224)(12 218)(13 219)(14 220)(15 27)(16 28)(17 22)(18 23)(19 24)(20 25)(21 26)(29 45)(30 46)(31 47)(32 48)(33 49)(34 43)(35 44)(50 74)(51 75)(52 76)(53 77)(54 71)(55 72)(56 73)(57 65)(58 66)(59 67)(60 68)(61 69)(62 70)(63 64)(78 102)(79 103)(80 104)(81 105)(82 99)(83 100)(84 101)(85 93)(86 94)(87 95)(88 96)(89 97)(90 98)(91 92)(106 130)(107 131)(108 132)(109 133)(110 127)(111 128)(112 129)(113 121)(114 122)(115 123)(116 124)(117 125)(118 126)(119 120)(134 151)(135 152)(136 153)(137 154)(138 148)(139 149)(140 150)(141 156)(142 157)(143 158)(144 159)(145 160)(146 161)(147 155)(162 179)(163 180)(164 181)(165 182)(166 176)(167 177)(168 178)(169 184)(170 185)(171 186)(172 187)(173 188)(174 189)(175 183)(190 207)(191 208)(192 209)(193 210)(194 204)(195 205)(196 206)(197 212)(198 213)(199 214)(200 215)(201 216)(202 217)(203 211)
(1 38)(2 39)(3 40)(4 41)(5 42)(6 36)(7 37)(8 27)(9 28)(10 22)(11 23)(12 24)(13 25)(14 26)(15 221)(16 222)(17 223)(18 224)(19 218)(20 219)(21 220)(29 45)(30 46)(31 47)(32 48)(33 49)(34 43)(35 44)(50 74)(51 75)(52 76)(53 77)(54 71)(55 72)(56 73)(57 65)(58 66)(59 67)(60 68)(61 69)(62 70)(63 64)(78 102)(79 103)(80 104)(81 105)(82 99)(83 100)(84 101)(85 93)(86 94)(87 95)(88 96)(89 97)(90 98)(91 92)(106 130)(107 131)(108 132)(109 133)(110 127)(111 128)(112 129)(113 121)(114 122)(115 123)(116 124)(117 125)(118 126)(119 120)(134 158)(135 159)(136 160)(137 161)(138 155)(139 156)(140 157)(141 149)(142 150)(143 151)(144 152)(145 153)(146 154)(147 148)(162 186)(163 187)(164 188)(165 189)(166 183)(167 184)(168 185)(169 177)(170 178)(171 179)(172 180)(173 181)(174 182)(175 176)(190 214)(191 215)(192 216)(193 217)(194 211)(195 212)(196 213)(197 205)(198 206)(199 207)(200 208)(201 209)(202 210)(203 204)
(1 47)(2 48)(3 49)(4 43)(5 44)(6 45)(7 46)(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(22 223)(23 224)(24 218)(25 219)(26 220)(27 221)(28 222)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(50 59)(51 60)(52 61)(53 62)(54 63)(55 57)(56 58)(64 71)(65 72)(66 73)(67 74)(68 75)(69 76)(70 77)(78 87)(79 88)(80 89)(81 90)(82 91)(83 85)(84 86)(92 99)(93 100)(94 101)(95 102)(96 103)(97 104)(98 105)(106 115)(107 116)(108 117)(109 118)(110 119)(111 113)(112 114)(120 127)(121 128)(122 129)(123 130)(124 131)(125 132)(126 133)(134 143)(135 144)(136 145)(137 146)(138 147)(139 141)(140 142)(148 155)(149 156)(150 157)(151 158)(152 159)(153 160)(154 161)(162 171)(163 172)(164 173)(165 174)(166 175)(167 169)(168 170)(176 183)(177 184)(178 185)(179 186)(180 187)(181 188)(182 189)(190 199)(191 200)(192 201)(193 202)(194 203)(195 197)(196 198)(204 211)(205 212)(206 213)(207 214)(208 215)(209 216)(210 217)
(1 158)(2 159)(3 160)(4 161)(5 155)(6 156)(7 157)(8 126)(9 120)(10 121)(11 122)(12 123)(13 124)(14 125)(15 133)(16 127)(17 128)(18 129)(19 130)(20 131)(21 132)(22 113)(23 114)(24 115)(25 116)(26 117)(27 118)(28 119)(29 141)(30 142)(31 143)(32 144)(33 145)(34 146)(35 147)(36 139)(37 140)(38 134)(39 135)(40 136)(41 137)(42 138)(43 154)(44 148)(45 149)(46 150)(47 151)(48 152)(49 153)(50 162)(51 163)(52 164)(53 165)(54 166)(55 167)(56 168)(57 169)(58 170)(59 171)(60 172)(61 173)(62 174)(63 175)(64 176)(65 177)(66 178)(67 179)(68 180)(69 181)(70 182)(71 183)(72 184)(73 185)(74 186)(75 187)(76 188)(77 189)(78 190)(79 191)(80 192)(81 193)(82 194)(83 195)(84 196)(85 197)(86 198)(87 199)(88 200)(89 201)(90 202)(91 203)(92 204)(93 205)(94 206)(95 207)(96 208)(97 209)(98 210)(99 211)(100 212)(101 213)(102 214)(103 215)(104 216)(105 217)(106 218)(107 219)(108 220)(109 221)(110 222)(111 223)(112 224)
(1 95 47 102)(2 96 48 103)(3 97 49 104)(4 98 43 105)(5 92 44 99)(6 93 45 100)(7 94 46 101)(8 174 15 165)(9 175 16 166)(10 169 17 167)(11 170 18 168)(12 171 19 162)(13 172 20 163)(14 173 21 164)(22 177 223 184)(23 178 224 185)(24 179 218 186)(25 180 219 187)(26 181 220 188)(27 182 221 189)(28 176 222 183)(29 83 36 85)(30 84 37 86)(31 78 38 87)(32 79 39 88)(33 80 40 89)(34 81 41 90)(35 82 42 91)(50 115 59 106)(51 116 60 107)(52 117 61 108)(53 118 62 109)(54 119 63 110)(55 113 57 111)(56 114 58 112)(64 127 71 120)(65 128 72 121)(66 129 73 122)(67 130 74 123)(68 131 75 124)(69 132 76 125)(70 133 77 126)(134 207 143 214)(135 208 144 215)(136 209 145 216)(137 210 146 217)(138 204 147 211)(139 205 141 212)(140 206 142 213)(148 194 155 203)(149 195 156 197)(150 196 157 198)(151 190 158 199)(152 191 159 200)(153 192 160 201)(154 193 161 202)
(1 50 38 74)(2 51 39 75)(3 52 40 76)(4 53 41 77)(5 54 42 71)(6 55 36 72)(7 56 37 73)(8 210 27 202)(9 204 28 203)(10 205 22 197)(11 206 23 198)(12 207 24 199)(13 208 25 200)(14 209 26 201)(15 217 221 193)(16 211 222 194)(17 212 223 195)(18 213 224 196)(19 214 218 190)(20 215 219 191)(21 216 220 192)(29 65 45 57)(30 66 46 58)(31 67 47 59)(32 68 48 60)(33 69 49 61)(34 70 43 62)(35 64 44 63)(78 130 102 106)(79 131 103 107)(80 132 104 108)(81 133 105 109)(82 127 99 110)(83 128 100 111)(84 129 101 112)(85 121 93 113)(86 122 94 114)(87 123 95 115)(88 124 96 116)(89 125 97 117)(90 126 98 118)(91 120 92 119)(134 186 158 162)(135 187 159 163)(136 188 160 164)(137 189 161 165)(138 183 155 166)(139 184 156 167)(140 185 157 168)(141 177 149 169)(142 178 150 170)(143 179 151 171)(144 180 152 172)(145 181 153 173)(146 182 154 174)(147 176 148 175)

G:=sub<Sym(224)| (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28)(29,30,31,32,33,34,35)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49)(50,51,52,53,54,55,56)(57,58,59,60,61,62,63)(64,65,66,67,68,69,70)(71,72,73,74,75,76,77)(78,79,80,81,82,83,84)(85,86,87,88,89,90,91)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112)(113,114,115,116,117,118,119)(120,121,122,123,124,125,126)(127,128,129,130,131,132,133)(134,135,136,137,138,139,140)(141,142,143,144,145,146,147)(148,149,150,151,152,153,154)(155,156,157,158,159,160,161)(162,163,164,165,166,167,168)(169,170,171,172,173,174,175)(176,177,178,179,180,181,182)(183,184,185,186,187,188,189)(190,191,192,193,194,195,196)(197,198,199,200,201,202,203)(204,205,206,207,208,209,210)(211,212,213,214,215,216,217)(218,219,220,221,222,223,224), (1,38)(2,39)(3,40)(4,41)(5,42)(6,36)(7,37)(8,221)(9,222)(10,223)(11,224)(12,218)(13,219)(14,220)(15,27)(16,28)(17,22)(18,23)(19,24)(20,25)(21,26)(29,45)(30,46)(31,47)(32,48)(33,49)(34,43)(35,44)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,65)(58,66)(59,67)(60,68)(61,69)(62,70)(63,64)(78,102)(79,103)(80,104)(81,105)(82,99)(83,100)(84,101)(85,93)(86,94)(87,95)(88,96)(89,97)(90,98)(91,92)(106,130)(107,131)(108,132)(109,133)(110,127)(111,128)(112,129)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,120)(134,151)(135,152)(136,153)(137,154)(138,148)(139,149)(140,150)(141,156)(142,157)(143,158)(144,159)(145,160)(146,161)(147,155)(162,179)(163,180)(164,181)(165,182)(166,176)(167,177)(168,178)(169,184)(170,185)(171,186)(172,187)(173,188)(174,189)(175,183)(190,207)(191,208)(192,209)(193,210)(194,204)(195,205)(196,206)(197,212)(198,213)(199,214)(200,215)(201,216)(202,217)(203,211), (1,38)(2,39)(3,40)(4,41)(5,42)(6,36)(7,37)(8,27)(9,28)(10,22)(11,23)(12,24)(13,25)(14,26)(15,221)(16,222)(17,223)(18,224)(19,218)(20,219)(21,220)(29,45)(30,46)(31,47)(32,48)(33,49)(34,43)(35,44)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,65)(58,66)(59,67)(60,68)(61,69)(62,70)(63,64)(78,102)(79,103)(80,104)(81,105)(82,99)(83,100)(84,101)(85,93)(86,94)(87,95)(88,96)(89,97)(90,98)(91,92)(106,130)(107,131)(108,132)(109,133)(110,127)(111,128)(112,129)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,120)(134,158)(135,159)(136,160)(137,161)(138,155)(139,156)(140,157)(141,149)(142,150)(143,151)(144,152)(145,153)(146,154)(147,148)(162,186)(163,187)(164,188)(165,189)(166,183)(167,184)(168,185)(169,177)(170,178)(171,179)(172,180)(173,181)(174,182)(175,176)(190,214)(191,215)(192,216)(193,217)(194,211)(195,212)(196,213)(197,205)(198,206)(199,207)(200,208)(201,209)(202,210)(203,204), (1,47)(2,48)(3,49)(4,43)(5,44)(6,45)(7,46)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(22,223)(23,224)(24,218)(25,219)(26,220)(27,221)(28,222)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(50,59)(51,60)(52,61)(53,62)(54,63)(55,57)(56,58)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77)(78,87)(79,88)(80,89)(81,90)(82,91)(83,85)(84,86)(92,99)(93,100)(94,101)(95,102)(96,103)(97,104)(98,105)(106,115)(107,116)(108,117)(109,118)(110,119)(111,113)(112,114)(120,127)(121,128)(122,129)(123,130)(124,131)(125,132)(126,133)(134,143)(135,144)(136,145)(137,146)(138,147)(139,141)(140,142)(148,155)(149,156)(150,157)(151,158)(152,159)(153,160)(154,161)(162,171)(163,172)(164,173)(165,174)(166,175)(167,169)(168,170)(176,183)(177,184)(178,185)(179,186)(180,187)(181,188)(182,189)(190,199)(191,200)(192,201)(193,202)(194,203)(195,197)(196,198)(204,211)(205,212)(206,213)(207,214)(208,215)(209,216)(210,217), (1,158)(2,159)(3,160)(4,161)(5,155)(6,156)(7,157)(8,126)(9,120)(10,121)(11,122)(12,123)(13,124)(14,125)(15,133)(16,127)(17,128)(18,129)(19,130)(20,131)(21,132)(22,113)(23,114)(24,115)(25,116)(26,117)(27,118)(28,119)(29,141)(30,142)(31,143)(32,144)(33,145)(34,146)(35,147)(36,139)(37,140)(38,134)(39,135)(40,136)(41,137)(42,138)(43,154)(44,148)(45,149)(46,150)(47,151)(48,152)(49,153)(50,162)(51,163)(52,164)(53,165)(54,166)(55,167)(56,168)(57,169)(58,170)(59,171)(60,172)(61,173)(62,174)(63,175)(64,176)(65,177)(66,178)(67,179)(68,180)(69,181)(70,182)(71,183)(72,184)(73,185)(74,186)(75,187)(76,188)(77,189)(78,190)(79,191)(80,192)(81,193)(82,194)(83,195)(84,196)(85,197)(86,198)(87,199)(88,200)(89,201)(90,202)(91,203)(92,204)(93,205)(94,206)(95,207)(96,208)(97,209)(98,210)(99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(106,218)(107,219)(108,220)(109,221)(110,222)(111,223)(112,224), (1,95,47,102)(2,96,48,103)(3,97,49,104)(4,98,43,105)(5,92,44,99)(6,93,45,100)(7,94,46,101)(8,174,15,165)(9,175,16,166)(10,169,17,167)(11,170,18,168)(12,171,19,162)(13,172,20,163)(14,173,21,164)(22,177,223,184)(23,178,224,185)(24,179,218,186)(25,180,219,187)(26,181,220,188)(27,182,221,189)(28,176,222,183)(29,83,36,85)(30,84,37,86)(31,78,38,87)(32,79,39,88)(33,80,40,89)(34,81,41,90)(35,82,42,91)(50,115,59,106)(51,116,60,107)(52,117,61,108)(53,118,62,109)(54,119,63,110)(55,113,57,111)(56,114,58,112)(64,127,71,120)(65,128,72,121)(66,129,73,122)(67,130,74,123)(68,131,75,124)(69,132,76,125)(70,133,77,126)(134,207,143,214)(135,208,144,215)(136,209,145,216)(137,210,146,217)(138,204,147,211)(139,205,141,212)(140,206,142,213)(148,194,155,203)(149,195,156,197)(150,196,157,198)(151,190,158,199)(152,191,159,200)(153,192,160,201)(154,193,161,202), (1,50,38,74)(2,51,39,75)(3,52,40,76)(4,53,41,77)(5,54,42,71)(6,55,36,72)(7,56,37,73)(8,210,27,202)(9,204,28,203)(10,205,22,197)(11,206,23,198)(12,207,24,199)(13,208,25,200)(14,209,26,201)(15,217,221,193)(16,211,222,194)(17,212,223,195)(18,213,224,196)(19,214,218,190)(20,215,219,191)(21,216,220,192)(29,65,45,57)(30,66,46,58)(31,67,47,59)(32,68,48,60)(33,69,49,61)(34,70,43,62)(35,64,44,63)(78,130,102,106)(79,131,103,107)(80,132,104,108)(81,133,105,109)(82,127,99,110)(83,128,100,111)(84,129,101,112)(85,121,93,113)(86,122,94,114)(87,123,95,115)(88,124,96,116)(89,125,97,117)(90,126,98,118)(91,120,92,119)(134,186,158,162)(135,187,159,163)(136,188,160,164)(137,189,161,165)(138,183,155,166)(139,184,156,167)(140,185,157,168)(141,177,149,169)(142,178,150,170)(143,179,151,171)(144,180,152,172)(145,181,153,173)(146,182,154,174)(147,176,148,175)>;

G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28)(29,30,31,32,33,34,35)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49)(50,51,52,53,54,55,56)(57,58,59,60,61,62,63)(64,65,66,67,68,69,70)(71,72,73,74,75,76,77)(78,79,80,81,82,83,84)(85,86,87,88,89,90,91)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112)(113,114,115,116,117,118,119)(120,121,122,123,124,125,126)(127,128,129,130,131,132,133)(134,135,136,137,138,139,140)(141,142,143,144,145,146,147)(148,149,150,151,152,153,154)(155,156,157,158,159,160,161)(162,163,164,165,166,167,168)(169,170,171,172,173,174,175)(176,177,178,179,180,181,182)(183,184,185,186,187,188,189)(190,191,192,193,194,195,196)(197,198,199,200,201,202,203)(204,205,206,207,208,209,210)(211,212,213,214,215,216,217)(218,219,220,221,222,223,224), (1,38)(2,39)(3,40)(4,41)(5,42)(6,36)(7,37)(8,221)(9,222)(10,223)(11,224)(12,218)(13,219)(14,220)(15,27)(16,28)(17,22)(18,23)(19,24)(20,25)(21,26)(29,45)(30,46)(31,47)(32,48)(33,49)(34,43)(35,44)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,65)(58,66)(59,67)(60,68)(61,69)(62,70)(63,64)(78,102)(79,103)(80,104)(81,105)(82,99)(83,100)(84,101)(85,93)(86,94)(87,95)(88,96)(89,97)(90,98)(91,92)(106,130)(107,131)(108,132)(109,133)(110,127)(111,128)(112,129)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,120)(134,151)(135,152)(136,153)(137,154)(138,148)(139,149)(140,150)(141,156)(142,157)(143,158)(144,159)(145,160)(146,161)(147,155)(162,179)(163,180)(164,181)(165,182)(166,176)(167,177)(168,178)(169,184)(170,185)(171,186)(172,187)(173,188)(174,189)(175,183)(190,207)(191,208)(192,209)(193,210)(194,204)(195,205)(196,206)(197,212)(198,213)(199,214)(200,215)(201,216)(202,217)(203,211), (1,38)(2,39)(3,40)(4,41)(5,42)(6,36)(7,37)(8,27)(9,28)(10,22)(11,23)(12,24)(13,25)(14,26)(15,221)(16,222)(17,223)(18,224)(19,218)(20,219)(21,220)(29,45)(30,46)(31,47)(32,48)(33,49)(34,43)(35,44)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,65)(58,66)(59,67)(60,68)(61,69)(62,70)(63,64)(78,102)(79,103)(80,104)(81,105)(82,99)(83,100)(84,101)(85,93)(86,94)(87,95)(88,96)(89,97)(90,98)(91,92)(106,130)(107,131)(108,132)(109,133)(110,127)(111,128)(112,129)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,120)(134,158)(135,159)(136,160)(137,161)(138,155)(139,156)(140,157)(141,149)(142,150)(143,151)(144,152)(145,153)(146,154)(147,148)(162,186)(163,187)(164,188)(165,189)(166,183)(167,184)(168,185)(169,177)(170,178)(171,179)(172,180)(173,181)(174,182)(175,176)(190,214)(191,215)(192,216)(193,217)(194,211)(195,212)(196,213)(197,205)(198,206)(199,207)(200,208)(201,209)(202,210)(203,204), (1,47)(2,48)(3,49)(4,43)(5,44)(6,45)(7,46)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(22,223)(23,224)(24,218)(25,219)(26,220)(27,221)(28,222)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(50,59)(51,60)(52,61)(53,62)(54,63)(55,57)(56,58)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77)(78,87)(79,88)(80,89)(81,90)(82,91)(83,85)(84,86)(92,99)(93,100)(94,101)(95,102)(96,103)(97,104)(98,105)(106,115)(107,116)(108,117)(109,118)(110,119)(111,113)(112,114)(120,127)(121,128)(122,129)(123,130)(124,131)(125,132)(126,133)(134,143)(135,144)(136,145)(137,146)(138,147)(139,141)(140,142)(148,155)(149,156)(150,157)(151,158)(152,159)(153,160)(154,161)(162,171)(163,172)(164,173)(165,174)(166,175)(167,169)(168,170)(176,183)(177,184)(178,185)(179,186)(180,187)(181,188)(182,189)(190,199)(191,200)(192,201)(193,202)(194,203)(195,197)(196,198)(204,211)(205,212)(206,213)(207,214)(208,215)(209,216)(210,217), (1,158)(2,159)(3,160)(4,161)(5,155)(6,156)(7,157)(8,126)(9,120)(10,121)(11,122)(12,123)(13,124)(14,125)(15,133)(16,127)(17,128)(18,129)(19,130)(20,131)(21,132)(22,113)(23,114)(24,115)(25,116)(26,117)(27,118)(28,119)(29,141)(30,142)(31,143)(32,144)(33,145)(34,146)(35,147)(36,139)(37,140)(38,134)(39,135)(40,136)(41,137)(42,138)(43,154)(44,148)(45,149)(46,150)(47,151)(48,152)(49,153)(50,162)(51,163)(52,164)(53,165)(54,166)(55,167)(56,168)(57,169)(58,170)(59,171)(60,172)(61,173)(62,174)(63,175)(64,176)(65,177)(66,178)(67,179)(68,180)(69,181)(70,182)(71,183)(72,184)(73,185)(74,186)(75,187)(76,188)(77,189)(78,190)(79,191)(80,192)(81,193)(82,194)(83,195)(84,196)(85,197)(86,198)(87,199)(88,200)(89,201)(90,202)(91,203)(92,204)(93,205)(94,206)(95,207)(96,208)(97,209)(98,210)(99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(106,218)(107,219)(108,220)(109,221)(110,222)(111,223)(112,224), (1,95,47,102)(2,96,48,103)(3,97,49,104)(4,98,43,105)(5,92,44,99)(6,93,45,100)(7,94,46,101)(8,174,15,165)(9,175,16,166)(10,169,17,167)(11,170,18,168)(12,171,19,162)(13,172,20,163)(14,173,21,164)(22,177,223,184)(23,178,224,185)(24,179,218,186)(25,180,219,187)(26,181,220,188)(27,182,221,189)(28,176,222,183)(29,83,36,85)(30,84,37,86)(31,78,38,87)(32,79,39,88)(33,80,40,89)(34,81,41,90)(35,82,42,91)(50,115,59,106)(51,116,60,107)(52,117,61,108)(53,118,62,109)(54,119,63,110)(55,113,57,111)(56,114,58,112)(64,127,71,120)(65,128,72,121)(66,129,73,122)(67,130,74,123)(68,131,75,124)(69,132,76,125)(70,133,77,126)(134,207,143,214)(135,208,144,215)(136,209,145,216)(137,210,146,217)(138,204,147,211)(139,205,141,212)(140,206,142,213)(148,194,155,203)(149,195,156,197)(150,196,157,198)(151,190,158,199)(152,191,159,200)(153,192,160,201)(154,193,161,202), (1,50,38,74)(2,51,39,75)(3,52,40,76)(4,53,41,77)(5,54,42,71)(6,55,36,72)(7,56,37,73)(8,210,27,202)(9,204,28,203)(10,205,22,197)(11,206,23,198)(12,207,24,199)(13,208,25,200)(14,209,26,201)(15,217,221,193)(16,211,222,194)(17,212,223,195)(18,213,224,196)(19,214,218,190)(20,215,219,191)(21,216,220,192)(29,65,45,57)(30,66,46,58)(31,67,47,59)(32,68,48,60)(33,69,49,61)(34,70,43,62)(35,64,44,63)(78,130,102,106)(79,131,103,107)(80,132,104,108)(81,133,105,109)(82,127,99,110)(83,128,100,111)(84,129,101,112)(85,121,93,113)(86,122,94,114)(87,123,95,115)(88,124,96,116)(89,125,97,117)(90,126,98,118)(91,120,92,119)(134,186,158,162)(135,187,159,163)(136,188,160,164)(137,189,161,165)(138,183,155,166)(139,184,156,167)(140,185,157,168)(141,177,149,169)(142,178,150,170)(143,179,151,171)(144,180,152,172)(145,181,153,173)(146,182,154,174)(147,176,148,175) );

G=PermutationGroup([[(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21),(22,23,24,25,26,27,28),(29,30,31,32,33,34,35),(36,37,38,39,40,41,42),(43,44,45,46,47,48,49),(50,51,52,53,54,55,56),(57,58,59,60,61,62,63),(64,65,66,67,68,69,70),(71,72,73,74,75,76,77),(78,79,80,81,82,83,84),(85,86,87,88,89,90,91),(92,93,94,95,96,97,98),(99,100,101,102,103,104,105),(106,107,108,109,110,111,112),(113,114,115,116,117,118,119),(120,121,122,123,124,125,126),(127,128,129,130,131,132,133),(134,135,136,137,138,139,140),(141,142,143,144,145,146,147),(148,149,150,151,152,153,154),(155,156,157,158,159,160,161),(162,163,164,165,166,167,168),(169,170,171,172,173,174,175),(176,177,178,179,180,181,182),(183,184,185,186,187,188,189),(190,191,192,193,194,195,196),(197,198,199,200,201,202,203),(204,205,206,207,208,209,210),(211,212,213,214,215,216,217),(218,219,220,221,222,223,224)], [(1,38),(2,39),(3,40),(4,41),(5,42),(6,36),(7,37),(8,221),(9,222),(10,223),(11,224),(12,218),(13,219),(14,220),(15,27),(16,28),(17,22),(18,23),(19,24),(20,25),(21,26),(29,45),(30,46),(31,47),(32,48),(33,49),(34,43),(35,44),(50,74),(51,75),(52,76),(53,77),(54,71),(55,72),(56,73),(57,65),(58,66),(59,67),(60,68),(61,69),(62,70),(63,64),(78,102),(79,103),(80,104),(81,105),(82,99),(83,100),(84,101),(85,93),(86,94),(87,95),(88,96),(89,97),(90,98),(91,92),(106,130),(107,131),(108,132),(109,133),(110,127),(111,128),(112,129),(113,121),(114,122),(115,123),(116,124),(117,125),(118,126),(119,120),(134,151),(135,152),(136,153),(137,154),(138,148),(139,149),(140,150),(141,156),(142,157),(143,158),(144,159),(145,160),(146,161),(147,155),(162,179),(163,180),(164,181),(165,182),(166,176),(167,177),(168,178),(169,184),(170,185),(171,186),(172,187),(173,188),(174,189),(175,183),(190,207),(191,208),(192,209),(193,210),(194,204),(195,205),(196,206),(197,212),(198,213),(199,214),(200,215),(201,216),(202,217),(203,211)], [(1,38),(2,39),(3,40),(4,41),(5,42),(6,36),(7,37),(8,27),(9,28),(10,22),(11,23),(12,24),(13,25),(14,26),(15,221),(16,222),(17,223),(18,224),(19,218),(20,219),(21,220),(29,45),(30,46),(31,47),(32,48),(33,49),(34,43),(35,44),(50,74),(51,75),(52,76),(53,77),(54,71),(55,72),(56,73),(57,65),(58,66),(59,67),(60,68),(61,69),(62,70),(63,64),(78,102),(79,103),(80,104),(81,105),(82,99),(83,100),(84,101),(85,93),(86,94),(87,95),(88,96),(89,97),(90,98),(91,92),(106,130),(107,131),(108,132),(109,133),(110,127),(111,128),(112,129),(113,121),(114,122),(115,123),(116,124),(117,125),(118,126),(119,120),(134,158),(135,159),(136,160),(137,161),(138,155),(139,156),(140,157),(141,149),(142,150),(143,151),(144,152),(145,153),(146,154),(147,148),(162,186),(163,187),(164,188),(165,189),(166,183),(167,184),(168,185),(169,177),(170,178),(171,179),(172,180),(173,181),(174,182),(175,176),(190,214),(191,215),(192,216),(193,217),(194,211),(195,212),(196,213),(197,205),(198,206),(199,207),(200,208),(201,209),(202,210),(203,204)], [(1,47),(2,48),(3,49),(4,43),(5,44),(6,45),(7,46),(8,15),(9,16),(10,17),(11,18),(12,19),(13,20),(14,21),(22,223),(23,224),(24,218),(25,219),(26,220),(27,221),(28,222),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(50,59),(51,60),(52,61),(53,62),(54,63),(55,57),(56,58),(64,71),(65,72),(66,73),(67,74),(68,75),(69,76),(70,77),(78,87),(79,88),(80,89),(81,90),(82,91),(83,85),(84,86),(92,99),(93,100),(94,101),(95,102),(96,103),(97,104),(98,105),(106,115),(107,116),(108,117),(109,118),(110,119),(111,113),(112,114),(120,127),(121,128),(122,129),(123,130),(124,131),(125,132),(126,133),(134,143),(135,144),(136,145),(137,146),(138,147),(139,141),(140,142),(148,155),(149,156),(150,157),(151,158),(152,159),(153,160),(154,161),(162,171),(163,172),(164,173),(165,174),(166,175),(167,169),(168,170),(176,183),(177,184),(178,185),(179,186),(180,187),(181,188),(182,189),(190,199),(191,200),(192,201),(193,202),(194,203),(195,197),(196,198),(204,211),(205,212),(206,213),(207,214),(208,215),(209,216),(210,217)], [(1,158),(2,159),(3,160),(4,161),(5,155),(6,156),(7,157),(8,126),(9,120),(10,121),(11,122),(12,123),(13,124),(14,125),(15,133),(16,127),(17,128),(18,129),(19,130),(20,131),(21,132),(22,113),(23,114),(24,115),(25,116),(26,117),(27,118),(28,119),(29,141),(30,142),(31,143),(32,144),(33,145),(34,146),(35,147),(36,139),(37,140),(38,134),(39,135),(40,136),(41,137),(42,138),(43,154),(44,148),(45,149),(46,150),(47,151),(48,152),(49,153),(50,162),(51,163),(52,164),(53,165),(54,166),(55,167),(56,168),(57,169),(58,170),(59,171),(60,172),(61,173),(62,174),(63,175),(64,176),(65,177),(66,178),(67,179),(68,180),(69,181),(70,182),(71,183),(72,184),(73,185),(74,186),(75,187),(76,188),(77,189),(78,190),(79,191),(80,192),(81,193),(82,194),(83,195),(84,196),(85,197),(86,198),(87,199),(88,200),(89,201),(90,202),(91,203),(92,204),(93,205),(94,206),(95,207),(96,208),(97,209),(98,210),(99,211),(100,212),(101,213),(102,214),(103,215),(104,216),(105,217),(106,218),(107,219),(108,220),(109,221),(110,222),(111,223),(112,224)], [(1,95,47,102),(2,96,48,103),(3,97,49,104),(4,98,43,105),(5,92,44,99),(6,93,45,100),(7,94,46,101),(8,174,15,165),(9,175,16,166),(10,169,17,167),(11,170,18,168),(12,171,19,162),(13,172,20,163),(14,173,21,164),(22,177,223,184),(23,178,224,185),(24,179,218,186),(25,180,219,187),(26,181,220,188),(27,182,221,189),(28,176,222,183),(29,83,36,85),(30,84,37,86),(31,78,38,87),(32,79,39,88),(33,80,40,89),(34,81,41,90),(35,82,42,91),(50,115,59,106),(51,116,60,107),(52,117,61,108),(53,118,62,109),(54,119,63,110),(55,113,57,111),(56,114,58,112),(64,127,71,120),(65,128,72,121),(66,129,73,122),(67,130,74,123),(68,131,75,124),(69,132,76,125),(70,133,77,126),(134,207,143,214),(135,208,144,215),(136,209,145,216),(137,210,146,217),(138,204,147,211),(139,205,141,212),(140,206,142,213),(148,194,155,203),(149,195,156,197),(150,196,157,198),(151,190,158,199),(152,191,159,200),(153,192,160,201),(154,193,161,202)], [(1,50,38,74),(2,51,39,75),(3,52,40,76),(4,53,41,77),(5,54,42,71),(6,55,36,72),(7,56,37,73),(8,210,27,202),(9,204,28,203),(10,205,22,197),(11,206,23,198),(12,207,24,199),(13,208,25,200),(14,209,26,201),(15,217,221,193),(16,211,222,194),(17,212,223,195),(18,213,224,196),(19,214,218,190),(20,215,219,191),(21,216,220,192),(29,65,45,57),(30,66,46,58),(31,67,47,59),(32,68,48,60),(33,69,49,61),(34,70,43,62),(35,64,44,63),(78,130,102,106),(79,131,103,107),(80,132,104,108),(81,133,105,109),(82,127,99,110),(83,128,100,111),(84,129,101,112),(85,121,93,113),(86,122,94,114),(87,123,95,115),(88,124,96,116),(89,125,97,117),(90,126,98,118),(91,120,92,119),(134,186,158,162),(135,187,159,163),(136,188,160,164),(137,189,161,165),(138,183,155,166),(139,184,156,167),(140,185,157,168),(141,177,149,169),(142,178,150,170),(143,179,151,171),(144,180,152,172),(145,181,153,173),(146,182,154,174),(147,176,148,175)]])

196 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4N 4O ··· 4T 7A ··· 7F 14A ··· 14R 14S ··· 14AD 14AE ··· 14AP 28A ··· 28X 28Y ··· 28CF 28CG ··· 28DP order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 ··· 4 7 ··· 7 14 ··· 14 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 28 ··· 28 size 1 1 1 1 2 2 4 4 1 1 1 1 2 ··· 2 4 ··· 4 1 ··· 1 1 ··· 1 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4

196 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C7 C14 C14 C14 C14 C14 C14 C14 C14 C14 C14 C4○D4 C4○D4 C7×C4○D4 C7×C4○D4 kernel C7×C23.36C23 C2×C4×C28 C7×C42⋊C2 D4×C28 Q8×C28 C7×C4⋊D4 C7×C22⋊Q8 C7×C22.D4 C7×C4.4D4 C7×C42.C2 C7×C42⋊2C2 C23.36C23 C2×C42 C42⋊C2 C4×D4 C4×Q8 C4⋊D4 C22⋊Q8 C22.D4 C4.4D4 C42.C2 C42⋊2C2 C28 C2×C14 C4 C22 # reps 1 1 2 3 1 1 1 2 1 1 2 6 6 12 18 6 6 6 12 6 6 12 8 4 48 24

Matrix representation of C7×C23.36C23 in GL4(𝔽29) generated by

 16 0 0 0 0 16 0 0 0 0 24 0 0 0 0 24
,
 28 8 0 0 0 1 0 0 0 0 1 0 0 0 28 28
,
 28 0 0 0 0 28 0 0 0 0 1 0 0 0 0 1
,
 28 0 0 0 0 28 0 0 0 0 28 0 0 0 0 28
,
 21 17 0 0 27 8 0 0 0 0 28 27 0 0 0 1
,
 17 9 0 0 0 12 0 0 0 0 12 0 0 0 0 12
,
 12 0 0 0 0 12 0 0 0 0 28 0 0 0 0 28
G:=sub<GL(4,GF(29))| [16,0,0,0,0,16,0,0,0,0,24,0,0,0,0,24],[28,0,0,0,8,1,0,0,0,0,1,28,0,0,0,28],[28,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[21,27,0,0,17,8,0,0,0,0,28,0,0,0,27,1],[17,0,0,0,9,12,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,28,0,0,0,0,28] >;

C7×C23.36C23 in GAP, Magma, Sage, TeX

C_7\times C_2^3._{36}C_2^3
% in TeX

G:=Group("C7xC2^3.36C2^3");
// GroupNames label

G:=SmallGroup(448,1312);
// by ID

G=gap.SmallGroup(448,1312);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1597,1576,4790,416]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^7=b^2=c^2=d^2=e^2=1,f^2=d,g^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,e*b*e=b*d=d*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f^-1=c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,e*g=g*e,f*g=g*f>;
// generators/relations

׿
×
𝔽