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## G = C7×C23⋊Q8order 448 = 26·7

### Direct product of C7 and C23⋊Q8

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

Series: Derived Chief Lower central Upper central

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

Generators and relations for C7×C23⋊Q8
G = < a,b,c,d,e,f | a7=b2=c2=d2=e4=1, f2=e2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf-1=bc=cb, ebe-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 362 in 202 conjugacy classes, 78 normal (14 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, Q8, C23, C23, C23, C14, C14, C14, C22⋊C4, C22×C4, C2×Q8, C24, C28, C2×C14, C2×C14, C2×C14, C2.C42, C2×C22⋊C4, C22×Q8, C2×C28, C2×C28, C7×Q8, C22×C14, C22×C14, C22×C14, C23⋊Q8, C7×C22⋊C4, C22×C28, Q8×C14, C23×C14, C7×C2.C42, C14×C22⋊C4, Q8×C2×C14, C7×C23⋊Q8
Quotients: C1, C2, C22, C7, D4, Q8, C23, C14, C2×D4, C2×Q8, C4○D4, C2×C14, C22≀C2, C22⋊Q8, C4.4D4, C7×D4, C7×Q8, C22×C14, C23⋊Q8, D4×C14, Q8×C14, C7×C4○D4, C7×C22≀C2, C7×C22⋊Q8, C7×C4.4D4, C7×C23⋊Q8

Smallest permutation representation of C7×C23⋊Q8
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 45)(9 46)(10 47)(11 48)(12 49)(13 43)(14 44)(15 42)(16 36)(17 37)(18 38)(19 39)(20 40)(21 41)(22 31)(23 32)(24 33)(25 34)(26 35)(27 29)(28 30)(99 153)(100 154)(101 148)(102 149)(103 150)(104 151)(105 152)(106 115)(107 116)(108 117)(109 118)(110 119)(111 113)(112 114)(120 129)(121 130)(122 131)(123 132)(124 133)(125 127)(126 128)(134 143)(135 144)(136 145)(137 146)(138 147)(139 141)(140 142)(155 173)(156 174)(157 175)(158 169)(159 170)(160 171)(161 172)(162 207)(163 208)(164 209)(165 210)(166 204)(167 205)(168 206)(176 194)(177 195)(178 196)(179 190)(180 191)(181 192)(182 193)(183 201)(184 202)(185 203)(186 197)(187 198)(188 199)(189 200)(211 220)(212 221)(213 222)(214 223)(215 224)(216 218)(217 219)
(1 76)(2 77)(3 71)(4 72)(5 73)(6 74)(7 75)(8 213)(9 214)(10 215)(11 216)(12 217)(13 211)(14 212)(15 33)(16 34)(17 35)(18 29)(19 30)(20 31)(21 32)(22 40)(23 41)(24 42)(25 36)(26 37)(27 38)(28 39)(43 220)(44 221)(45 222)(46 223)(47 224)(48 218)(49 219)(50 95)(51 96)(52 97)(53 98)(54 92)(55 93)(56 94)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 82)(65 83)(66 84)(67 78)(68 79)(69 80)(70 81)(99 117)(100 118)(101 119)(102 113)(103 114)(104 115)(105 116)(106 151)(107 152)(108 153)(109 154)(110 148)(111 149)(112 150)(120 138)(121 139)(122 140)(123 134)(124 135)(125 136)(126 137)(127 145)(128 146)(129 147)(130 141)(131 142)(132 143)(133 144)(155 173)(156 174)(157 175)(158 169)(159 170)(160 171)(161 172)(162 207)(163 208)(164 209)(165 210)(166 204)(167 205)(168 206)(176 194)(177 195)(178 196)(179 190)(180 191)(181 192)(182 193)(183 201)(184 202)(185 203)(186 197)(187 198)(188 199)(189 200)
(1 78)(2 79)(3 80)(4 81)(5 82)(6 83)(7 84)(8 222)(9 223)(10 224)(11 218)(12 219)(13 220)(14 221)(15 24)(16 25)(17 26)(18 27)(19 28)(20 22)(21 23)(29 38)(30 39)(31 40)(32 41)(33 42)(34 36)(35 37)(43 211)(44 212)(45 213)(46 214)(47 215)(48 216)(49 217)(50 59)(51 60)(52 61)(53 62)(54 63)(55 57)(56 58)(64 73)(65 74)(66 75)(67 76)(68 77)(69 71)(70 72)(85 93)(86 94)(87 95)(88 96)(89 97)(90 98)(91 92)(99 153)(100 154)(101 148)(102 149)(103 150)(104 151)(105 152)(106 115)(107 116)(108 117)(109 118)(110 119)(111 113)(112 114)(120 129)(121 130)(122 131)(123 132)(124 133)(125 127)(126 128)(134 143)(135 144)(136 145)(137 146)(138 147)(139 141)(140 142)(155 209)(156 210)(157 204)(158 205)(159 206)(160 207)(161 208)(162 171)(163 172)(164 173)(165 174)(166 175)(167 169)(168 170)(176 185)(177 186)(178 187)(179 188)(180 189)(181 183)(182 184)(190 199)(191 200)(192 201)(193 202)(194 203)(195 197)(196 198)
(1 115 59 143)(2 116 60 144)(3 117 61 145)(4 118 62 146)(5 119 63 147)(6 113 57 141)(7 114 58 142)(8 203 37 175)(9 197 38 169)(10 198 39 170)(11 199 40 171)(12 200 41 172)(13 201 42 173)(14 202 36 174)(15 209 43 181)(16 210 44 182)(17 204 45 176)(18 205 46 177)(19 206 47 178)(20 207 48 179)(21 208 49 180)(22 160 216 188)(23 161 217 189)(24 155 211 183)(25 156 212 184)(26 157 213 185)(27 158 214 186)(28 159 215 187)(29 167 223 195)(30 168 224 196)(31 162 218 190)(32 163 219 191)(33 164 220 192)(34 165 221 193)(35 166 222 194)(50 134 78 106)(51 135 79 107)(52 136 80 108)(53 137 81 109)(54 138 82 110)(55 139 83 111)(56 140 84 112)(64 148 92 120)(65 149 93 121)(66 150 94 122)(67 151 95 123)(68 152 96 124)(69 153 97 125)(70 154 98 126)(71 99 89 127)(72 100 90 128)(73 101 91 129)(74 102 85 130)(75 103 86 131)(76 104 87 132)(77 105 88 133)
(1 171 59 199)(2 172 60 200)(3 173 61 201)(4 174 62 202)(5 175 63 203)(6 169 57 197)(7 170 58 198)(8 119 37 147)(9 113 38 141)(10 114 39 142)(11 115 40 143)(12 116 41 144)(13 117 42 145)(14 118 36 146)(15 125 43 153)(16 126 44 154)(17 120 45 148)(18 121 46 149)(19 122 47 150)(20 123 48 151)(21 124 49 152)(22 132 216 104)(23 133 217 105)(24 127 211 99)(25 128 212 100)(26 129 213 101)(27 130 214 102)(28 131 215 103)(29 139 223 111)(30 140 224 112)(31 134 218 106)(32 135 219 107)(33 136 220 108)(34 137 221 109)(35 138 222 110)(50 190 78 162)(51 191 79 163)(52 192 80 164)(53 193 81 165)(54 194 82 166)(55 195 83 167)(56 196 84 168)(64 204 92 176)(65 205 93 177)(66 206 94 178)(67 207 95 179)(68 208 96 180)(69 209 97 181)(70 210 98 182)(71 155 89 183)(72 156 90 184)(73 157 91 185)(74 158 85 186)(75 159 86 187)(76 160 87 188)(77 161 88 189)

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

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

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

154 conjugacy classes

 class 1 2A ··· 2G 2H 2I 4A ··· 4L 7A ··· 7F 14A ··· 14AP 14AQ ··· 14BB 28A ··· 28BT order 1 2 ··· 2 2 2 4 ··· 4 7 ··· 7 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 ··· 1 4 4 4 ··· 4 1 ··· 1 1 ··· 1 4 ··· 4 4 ··· 4

154 irreducible representations

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

Matrix representation of C7×C23⋊Q8 in GL6(𝔽29)

 7 0 0 0 0 0 0 7 0 0 0 0 0 0 24 0 0 0 0 0 0 24 0 0 0 0 0 0 24 0 0 0 0 0 0 24
,
 1 0 0 0 0 0 0 28 0 0 0 0 0 0 1 0 0 0 0 0 5 28 0 0 0 0 0 0 1 0 0 0 0 0 0 28
,
 28 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 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 28 0 0 0 0 0 0 28
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 28 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 10 25 0 0 0 0 3 19 0 0 0 0 0 0 0 17 0 0 0 0 17 0

G:=sub<GL(6,GF(29))| [7,0,0,0,0,0,0,7,0,0,0,0,0,0,24,0,0,0,0,0,0,24,0,0,0,0,0,0,24,0,0,0,0,0,0,24],[1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,5,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,28,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,10,3,0,0,0,0,25,19,0,0,0,0,0,0,0,17,0,0,0,0,17,0] >;

C7×C23⋊Q8 in GAP, Magma, Sage, TeX

C_7\times C_2^3\rtimes Q_8
% in TeX

G:=Group("C7xC2^3:Q8");
// GroupNames label

G:=SmallGroup(448,801);
// by ID

G=gap.SmallGroup(448,801);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,392,813,400,2438,2403]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^7=b^2=c^2=d^2=e^4=1,f^2=e^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,f*b*f^-1=b*c=c*b,e*b*e^-1=b*d=d*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations

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