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## G = C7×C23.41C23order 448 = 26·7

### Direct product of C7 and C23.41C23

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C7×C23.41C23
 Chief series C1 — C2 — C22 — C2×C14 — C2×C28 — C7×C4⋊C4 — C7×C4⋊Q8 — C7×C23.41C23
 Lower central C1 — C22 — C7×C23.41C23
 Upper central C1 — C2×C14 — C7×C23.41C23

Generators and relations for C7×C23.41C23
G = < a,b,c,d,e,f,g | a7=b2=c2=d2=1, e2=g2=d, f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, ebe-1=bc=cb, bd=db, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, gfg-1=cf=fc, cg=gc, geg-1=de=ed, df=fd, dg=gd >

Subgroups: 274 in 206 conjugacy classes, 162 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, Q8, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C28, C28, C2×C14, C2×C14, C2×C14, C2×C4⋊C4, C42⋊C2, C22⋊Q8, C42.C2, C4⋊Q8, C2×C28, C2×C28, C7×Q8, C22×C14, C23.41C23, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C22×C28, C22×C28, Q8×C14, C14×C4⋊C4, C7×C42⋊C2, C7×C22⋊Q8, C7×C42.C2, C7×C4⋊Q8, C7×C23.41C23
Quotients: C1, C2, C22, C7, Q8, C23, C14, C2×Q8, C24, C2×C14, C22×Q8, 2+ 1+4, 2- 1+4, C7×Q8, C22×C14, C23.41C23, Q8×C14, C23×C14, Q8×C2×C14, C7×2+ 1+4, C7×2- 1+4, C7×C23.41C23

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

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

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

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

154 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E ··· 4P 7A ··· 7F 14A ··· 14R 14S ··· 14AD 28A ··· 28X 28Y ··· 28CR order 1 2 2 2 2 2 4 4 4 4 4 ··· 4 7 ··· 7 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 1 1 2 2 2 2 2 2 4 ··· 4 1 ··· 1 1 ··· 1 2 ··· 2 2 ··· 2 4 ··· 4

154 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 4 4 4 type + + + + + + - + - image C1 C2 C2 C2 C2 C2 C7 C14 C14 C14 C14 C14 Q8 C7×Q8 2+ 1+4 2- 1+4 C7×2+ 1+4 C7×2- 1+4 kernel C7×C23.41C23 C14×C4⋊C4 C7×C42⋊C2 C7×C22⋊Q8 C7×C42.C2 C7×C4⋊Q8 C23.41C23 C2×C4⋊C4 C42⋊C2 C22⋊Q8 C42.C2 C4⋊Q8 C2×C28 C2×C4 C14 C14 C2 C2 # reps 1 1 2 4 4 4 6 6 12 24 24 24 4 24 1 1 6 6

Matrix representation of C7×C23.41C23 in GL6(𝔽29)

 25 0 0 0 0 0 0 25 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 28 0 0 0 0 0 0 28 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 4 22 28 0 0 0 26 5 0 28
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28
,
 28 0 0 0 0 0 0 28 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 21 14 0 0 0 0 14 8 0 0 0 0 0 0 25 7 2 0 0 0 1 27 27 27 0 0 18 21 11 7 0 0 8 13 21 24
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 28 0 0 0 0 1 0 0 0 0 0 26 26 28 28 0 0 0 6 2 1
,
 0 28 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 26 1 28 28 0 0 4 25 0 1

G:=sub<GL(6,GF(29))| [25,0,0,0,0,0,0,25,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,4,26,0,0,0,1,22,5,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[21,14,0,0,0,0,14,8,0,0,0,0,0,0,25,1,18,8,0,0,7,27,21,13,0,0,2,27,11,21,0,0,0,27,7,24],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,26,0,0,0,28,0,26,6,0,0,0,0,28,2,0,0,0,0,28,1],[0,1,0,0,0,0,28,0,0,0,0,0,0,0,0,1,26,4,0,0,1,0,1,25,0,0,0,0,28,0,0,0,0,0,28,1] >;

C7×C23.41C23 in GAP, Magma, Sage, TeX

C_7\times C_2^3._{41}C_2^3
% in TeX

G:=Group("C7xC2^3.41C2^3");
// GroupNames label

G:=SmallGroup(448,1327);
// by ID

G=gap.SmallGroup(448,1327);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1568,1597,792,4790,1227,1192,3363]);
// Polycyclic

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

׿
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