metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C7⋊C8⋊3D4, C7⋊2(C8⋊D4), C4⋊C4.28D14, C4.167(D4×D7), D14⋊3Q8⋊2C2, Q8⋊C4⋊18D7, (C2×C8).177D14, C4⋊D28.3C2, C28.125(C2×D4), (C2×Q8).18D14, C2.D56⋊25C2, C28.20(C4○D4), C4.33(C4○D28), C28.Q8⋊10C2, (C2×Dic7).33D4, (C22×D7).19D4, C22.202(D4×D7), C14.24(C4⋊D4), C2.18(D56⋊C2), C14.64(C8⋊C22), (C2×C28).252C23, (C2×C56).201C22, (C2×D28).66C22, C4⋊Dic7.96C22, (Q8×C14).35C22, C2.27(D14⋊D4), C2.16(Q16⋊D7), C14.62(C8.C22), (C2×Q8⋊D7)⋊5C2, (C2×C8⋊D7)⋊20C2, (C2×C7⋊C8).43C22, (C2×C4×D7).25C22, (C7×Q8⋊C4)⋊23C2, (C2×C14).265(C2×D4), (C7×C4⋊C4).53C22, (C2×C4).359(C22×D7), SmallGroup(448,346)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7⋊(C8⋊D4)
G = < a,b,c,d | a7=b8=c4=d2=1, bab-1=dad=a-1, ac=ca, cbc-1=b-1, dbd=b3, dcd=c-1 >
Subgroups: 756 in 120 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), SD16, C22×C4, C2×D4, C2×Q8, Dic7, C28, C28, D14, C2×C14, D4⋊C4, Q8⋊C4, C2.D8, C4⋊D4, C22⋊Q8, C2×M4(2), C2×SD16, C7⋊C8, C56, C4×D7, D28, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×Q8, C22×D7, C22×D7, C8⋊D4, C8⋊D7, C2×C7⋊C8, Dic7⋊C4, C4⋊Dic7, D14⋊C4, Q8⋊D7, C7×C4⋊C4, C2×C56, C2×C4×D7, C2×D28, C2×D28, Q8×C14, C28.Q8, C2.D56, C7×Q8⋊C4, C4⋊D28, C2×C8⋊D7, C2×Q8⋊D7, D14⋊3Q8, C7⋊(C8⋊D4)
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C8⋊C22, C8.C22, C22×D7, C8⋊D4, C4○D28, D4×D7, D14⋊D4, D56⋊C2, Q16⋊D7, C7⋊(C8⋊D4)
►On 224 points
Generators in S224
(1 150 62 112 179 219 174)(2 175 220 180 105 63 151)(3 152 64 106 181 221 176)(4 169 222 182 107 57 145)(5 146 58 108 183 223 170)(6 171 224 184 109 59 147)(7 148 60 110 177 217 172)(8 173 218 178 111 61 149)(9 98 138 130 56 45 215)(10 216 46 49 131 139 99)(11 100 140 132 50 47 209)(12 210 48 51 133 141 101)(13 102 142 134 52 41 211)(14 212 42 53 135 143 103)(15 104 144 136 54 43 213)(16 214 44 55 129 137 97)(17 188 199 74 90 160 34)(18 35 153 91 75 200 189)(19 190 193 76 92 154 36)(20 37 155 93 77 194 191)(21 192 195 78 94 156 38)(22 39 157 95 79 196 185)(23 186 197 80 96 158 40)(24 33 159 89 73 198 187)(25 127 116 84 66 166 204)(26 205 167 67 85 117 128)(27 121 118 86 68 168 206)(28 207 161 69 87 119 122)(29 123 120 88 70 162 208)(30 201 163 71 81 113 124)(31 125 114 82 72 164 202)(32 203 165 65 83 115 126)
(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)
(1 157 97 164)(2 156 98 163)(3 155 99 162)(4 154 100 161)(5 153 101 168)(6 160 102 167)(7 159 103 166)(8 158 104 165)(9 201 151 94)(10 208 152 93)(11 207 145 92)(12 206 146 91)(13 205 147 90)(14 204 148 89)(15 203 149 96)(16 202 150 95)(17 134 85 224)(18 133 86 223)(19 132 87 222)(20 131 88 221)(21 130 81 220)(22 129 82 219)(23 136 83 218)(24 135 84 217)(25 60 73 212)(26 59 74 211)(27 58 75 210)(28 57 76 209)(29 64 77 216)(30 63 78 215)(31 62 79 214)(32 61 80 213)(33 143 66 172)(34 142 67 171)(35 141 68 170)(36 140 69 169)(37 139 70 176)(38 138 71 175)(39 137 72 174)(40 144 65 173)(41 128 109 199)(42 127 110 198)(43 126 111 197)(44 125 112 196)(45 124 105 195)(46 123 106 194)(47 122 107 193)(48 121 108 200)(49 120 181 191)(50 119 182 190)(51 118 183 189)(52 117 184 188)(53 116 177 187)(54 115 178 186)(55 114 179 185)(56 113 180 192)
(1 164)(2 167)(3 162)(4 165)(5 168)(6 163)(7 166)(8 161)(9 34)(10 37)(11 40)(12 35)(13 38)(14 33)(15 36)(16 39)(17 215)(18 210)(19 213)(20 216)(21 211)(22 214)(23 209)(24 212)(25 217)(26 220)(27 223)(28 218)(29 221)(30 224)(31 219)(32 222)(41 192)(42 187)(43 190)(44 185)(45 188)(46 191)(47 186)(48 189)(49 194)(50 197)(51 200)(52 195)(53 198)(54 193)(55 196)(56 199)(57 83)(58 86)(59 81)(60 84)(61 87)(62 82)(63 85)(64 88)(65 145)(66 148)(67 151)(68 146)(69 149)(70 152)(71 147)(72 150)(73 135)(74 130)(75 133)(76 136)(77 131)(78 134)(79 129)(80 132)(89 143)(90 138)(91 141)(92 144)(93 139)(94 142)(95 137)(96 140)(97 157)(98 160)(99 155)(100 158)(101 153)(102 156)(103 159)(104 154)(105 117)(106 120)(107 115)(108 118)(109 113)(110 116)(111 119)(112 114)(121 183)(122 178)(123 181)(124 184)(125 179)(126 182)(127 177)(128 180)(169 203)(170 206)(171 201)(172 204)(173 207)(174 202)(175 205)(176 208)
G:=sub<Sym(224)| (1,150,62,112,179,219,174)(2,175,220,180,105,63,151)(3,152,64,106,181,221,176)(4,169,222,182,107,57,145)(5,146,58,108,183,223,170)(6,171,224,184,109,59,147)(7,148,60,110,177,217,172)(8,173,218,178,111,61,149)(9,98,138,130,56,45,215)(10,216,46,49,131,139,99)(11,100,140,132,50,47,209)(12,210,48,51,133,141,101)(13,102,142,134,52,41,211)(14,212,42,53,135,143,103)(15,104,144,136,54,43,213)(16,214,44,55,129,137,97)(17,188,199,74,90,160,34)(18,35,153,91,75,200,189)(19,190,193,76,92,154,36)(20,37,155,93,77,194,191)(21,192,195,78,94,156,38)(22,39,157,95,79,196,185)(23,186,197,80,96,158,40)(24,33,159,89,73,198,187)(25,127,116,84,66,166,204)(26,205,167,67,85,117,128)(27,121,118,86,68,168,206)(28,207,161,69,87,119,122)(29,123,120,88,70,162,208)(30,201,163,71,81,113,124)(31,125,114,82,72,164,202)(32,203,165,65,83,115,126), (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), (1,157,97,164)(2,156,98,163)(3,155,99,162)(4,154,100,161)(5,153,101,168)(6,160,102,167)(7,159,103,166)(8,158,104,165)(9,201,151,94)(10,208,152,93)(11,207,145,92)(12,206,146,91)(13,205,147,90)(14,204,148,89)(15,203,149,96)(16,202,150,95)(17,134,85,224)(18,133,86,223)(19,132,87,222)(20,131,88,221)(21,130,81,220)(22,129,82,219)(23,136,83,218)(24,135,84,217)(25,60,73,212)(26,59,74,211)(27,58,75,210)(28,57,76,209)(29,64,77,216)(30,63,78,215)(31,62,79,214)(32,61,80,213)(33,143,66,172)(34,142,67,171)(35,141,68,170)(36,140,69,169)(37,139,70,176)(38,138,71,175)(39,137,72,174)(40,144,65,173)(41,128,109,199)(42,127,110,198)(43,126,111,197)(44,125,112,196)(45,124,105,195)(46,123,106,194)(47,122,107,193)(48,121,108,200)(49,120,181,191)(50,119,182,190)(51,118,183,189)(52,117,184,188)(53,116,177,187)(54,115,178,186)(55,114,179,185)(56,113,180,192), (1,164)(2,167)(3,162)(4,165)(5,168)(6,163)(7,166)(8,161)(9,34)(10,37)(11,40)(12,35)(13,38)(14,33)(15,36)(16,39)(17,215)(18,210)(19,213)(20,216)(21,211)(22,214)(23,209)(24,212)(25,217)(26,220)(27,223)(28,218)(29,221)(30,224)(31,219)(32,222)(41,192)(42,187)(43,190)(44,185)(45,188)(46,191)(47,186)(48,189)(49,194)(50,197)(51,200)(52,195)(53,198)(54,193)(55,196)(56,199)(57,83)(58,86)(59,81)(60,84)(61,87)(62,82)(63,85)(64,88)(65,145)(66,148)(67,151)(68,146)(69,149)(70,152)(71,147)(72,150)(73,135)(74,130)(75,133)(76,136)(77,131)(78,134)(79,129)(80,132)(89,143)(90,138)(91,141)(92,144)(93,139)(94,142)(95,137)(96,140)(97,157)(98,160)(99,155)(100,158)(101,153)(102,156)(103,159)(104,154)(105,117)(106,120)(107,115)(108,118)(109,113)(110,116)(111,119)(112,114)(121,183)(122,178)(123,181)(124,184)(125,179)(126,182)(127,177)(128,180)(169,203)(170,206)(171,201)(172,204)(173,207)(174,202)(175,205)(176,208)>;
G:=Group( (1,150,62,112,179,219,174)(2,175,220,180,105,63,151)(3,152,64,106,181,221,176)(4,169,222,182,107,57,145)(5,146,58,108,183,223,170)(6,171,224,184,109,59,147)(7,148,60,110,177,217,172)(8,173,218,178,111,61,149)(9,98,138,130,56,45,215)(10,216,46,49,131,139,99)(11,100,140,132,50,47,209)(12,210,48,51,133,141,101)(13,102,142,134,52,41,211)(14,212,42,53,135,143,103)(15,104,144,136,54,43,213)(16,214,44,55,129,137,97)(17,188,199,74,90,160,34)(18,35,153,91,75,200,189)(19,190,193,76,92,154,36)(20,37,155,93,77,194,191)(21,192,195,78,94,156,38)(22,39,157,95,79,196,185)(23,186,197,80,96,158,40)(24,33,159,89,73,198,187)(25,127,116,84,66,166,204)(26,205,167,67,85,117,128)(27,121,118,86,68,168,206)(28,207,161,69,87,119,122)(29,123,120,88,70,162,208)(30,201,163,71,81,113,124)(31,125,114,82,72,164,202)(32,203,165,65,83,115,126), (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), (1,157,97,164)(2,156,98,163)(3,155,99,162)(4,154,100,161)(5,153,101,168)(6,160,102,167)(7,159,103,166)(8,158,104,165)(9,201,151,94)(10,208,152,93)(11,207,145,92)(12,206,146,91)(13,205,147,90)(14,204,148,89)(15,203,149,96)(16,202,150,95)(17,134,85,224)(18,133,86,223)(19,132,87,222)(20,131,88,221)(21,130,81,220)(22,129,82,219)(23,136,83,218)(24,135,84,217)(25,60,73,212)(26,59,74,211)(27,58,75,210)(28,57,76,209)(29,64,77,216)(30,63,78,215)(31,62,79,214)(32,61,80,213)(33,143,66,172)(34,142,67,171)(35,141,68,170)(36,140,69,169)(37,139,70,176)(38,138,71,175)(39,137,72,174)(40,144,65,173)(41,128,109,199)(42,127,110,198)(43,126,111,197)(44,125,112,196)(45,124,105,195)(46,123,106,194)(47,122,107,193)(48,121,108,200)(49,120,181,191)(50,119,182,190)(51,118,183,189)(52,117,184,188)(53,116,177,187)(54,115,178,186)(55,114,179,185)(56,113,180,192), (1,164)(2,167)(3,162)(4,165)(5,168)(6,163)(7,166)(8,161)(9,34)(10,37)(11,40)(12,35)(13,38)(14,33)(15,36)(16,39)(17,215)(18,210)(19,213)(20,216)(21,211)(22,214)(23,209)(24,212)(25,217)(26,220)(27,223)(28,218)(29,221)(30,224)(31,219)(32,222)(41,192)(42,187)(43,190)(44,185)(45,188)(46,191)(47,186)(48,189)(49,194)(50,197)(51,200)(52,195)(53,198)(54,193)(55,196)(56,199)(57,83)(58,86)(59,81)(60,84)(61,87)(62,82)(63,85)(64,88)(65,145)(66,148)(67,151)(68,146)(69,149)(70,152)(71,147)(72,150)(73,135)(74,130)(75,133)(76,136)(77,131)(78,134)(79,129)(80,132)(89,143)(90,138)(91,141)(92,144)(93,139)(94,142)(95,137)(96,140)(97,157)(98,160)(99,155)(100,158)(101,153)(102,156)(103,159)(104,154)(105,117)(106,120)(107,115)(108,118)(109,113)(110,116)(111,119)(112,114)(121,183)(122,178)(123,181)(124,184)(125,179)(126,182)(127,177)(128,180)(169,203)(170,206)(171,201)(172,204)(173,207)(174,202)(175,205)(176,208) );
G=PermutationGroup([[(1,150,62,112,179,219,174),(2,175,220,180,105,63,151),(3,152,64,106,181,221,176),(4,169,222,182,107,57,145),(5,146,58,108,183,223,170),(6,171,224,184,109,59,147),(7,148,60,110,177,217,172),(8,173,218,178,111,61,149),(9,98,138,130,56,45,215),(10,216,46,49,131,139,99),(11,100,140,132,50,47,209),(12,210,48,51,133,141,101),(13,102,142,134,52,41,211),(14,212,42,53,135,143,103),(15,104,144,136,54,43,213),(16,214,44,55,129,137,97),(17,188,199,74,90,160,34),(18,35,153,91,75,200,189),(19,190,193,76,92,154,36),(20,37,155,93,77,194,191),(21,192,195,78,94,156,38),(22,39,157,95,79,196,185),(23,186,197,80,96,158,40),(24,33,159,89,73,198,187),(25,127,116,84,66,166,204),(26,205,167,67,85,117,128),(27,121,118,86,68,168,206),(28,207,161,69,87,119,122),(29,123,120,88,70,162,208),(30,201,163,71,81,113,124),(31,125,114,82,72,164,202),(32,203,165,65,83,115,126)], [(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)], [(1,157,97,164),(2,156,98,163),(3,155,99,162),(4,154,100,161),(5,153,101,168),(6,160,102,167),(7,159,103,166),(8,158,104,165),(9,201,151,94),(10,208,152,93),(11,207,145,92),(12,206,146,91),(13,205,147,90),(14,204,148,89),(15,203,149,96),(16,202,150,95),(17,134,85,224),(18,133,86,223),(19,132,87,222),(20,131,88,221),(21,130,81,220),(22,129,82,219),(23,136,83,218),(24,135,84,217),(25,60,73,212),(26,59,74,211),(27,58,75,210),(28,57,76,209),(29,64,77,216),(30,63,78,215),(31,62,79,214),(32,61,80,213),(33,143,66,172),(34,142,67,171),(35,141,68,170),(36,140,69,169),(37,139,70,176),(38,138,71,175),(39,137,72,174),(40,144,65,173),(41,128,109,199),(42,127,110,198),(43,126,111,197),(44,125,112,196),(45,124,105,195),(46,123,106,194),(47,122,107,193),(48,121,108,200),(49,120,181,191),(50,119,182,190),(51,118,183,189),(52,117,184,188),(53,116,177,187),(54,115,178,186),(55,114,179,185),(56,113,180,192)], [(1,164),(2,167),(3,162),(4,165),(5,168),(6,163),(7,166),(8,161),(9,34),(10,37),(11,40),(12,35),(13,38),(14,33),(15,36),(16,39),(17,215),(18,210),(19,213),(20,216),(21,211),(22,214),(23,209),(24,212),(25,217),(26,220),(27,223),(28,218),(29,221),(30,224),(31,219),(32,222),(41,192),(42,187),(43,190),(44,185),(45,188),(46,191),(47,186),(48,189),(49,194),(50,197),(51,200),(52,195),(53,198),(54,193),(55,196),(56,199),(57,83),(58,86),(59,81),(60,84),(61,87),(62,82),(63,85),(64,88),(65,145),(66,148),(67,151),(68,146),(69,149),(70,152),(71,147),(72,150),(73,135),(74,130),(75,133),(76,136),(77,131),(78,134),(79,129),(80,132),(89,143),(90,138),(91,141),(92,144),(93,139),(94,142),(95,137),(96,140),(97,157),(98,160),(99,155),(100,158),(101,153),(102,156),(103,159),(104,154),(105,117),(106,120),(107,115),(108,118),(109,113),(110,116),(111,119),(112,114),(121,183),(122,178),(123,181),(124,184),(125,179),(126,182),(127,177),(128,180),(169,203),(170,206),(171,201),(172,204),(173,207),(174,202),(175,205),(176,208)]])
58 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 28 | 56 | 2 | 2 | 8 | 8 | 28 | 56 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
58 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | C4○D4 | D14 | D14 | D14 | C4○D28 | C8⋊C22 | C8.C22 | D4×D7 | D4×D7 | D56⋊C2 | Q16⋊D7 |
| kernel | C7⋊(C8⋊D4) | C28.Q8 | C2.D56 | C7×Q8⋊C4 | C4⋊D28 | C2×C8⋊D7 | C2×Q8⋊D7 | D14⋊3Q8 | C7⋊C8 | C2×Dic7 | C22×D7 | Q8⋊C4 | C28 | C4⋊C4 | C2×C8 | C2×Q8 | C4 | C14 | C14 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 3 | 2 | 3 | 3 | 3 | 12 | 1 | 1 | 3 | 3 | 6 | 6 |
Matrix representation of C7⋊(C8⋊D4) ►in GL6(𝔽113)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 79 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 112 | 79 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 47 | 105 | 66 |
| 0 | 0 | 1 | 105 | 112 | 8 |
| 0 | 0 | 4 | 80 | 0 | 0 |
| 0 | 0 | 57 | 109 | 0 | 0 |
| 5 | 77 | 0 | 0 | 0 | 0 |
| 7 | 108 | 0 | 0 | 0 | 0 |
| 0 | 0 | 87 | 45 | 53 | 43 |
| 0 | 0 | 68 | 26 | 70 | 60 |
| 0 | 0 | 57 | 10 | 26 | 68 |
| 0 | 0 | 103 | 56 | 45 | 87 |
| 5 | 77 | 0 | 0 | 0 | 0 |
| 76 | 108 | 0 | 0 | 0 | 0 |
| 0 | 0 | 87 | 45 | 53 | 43 |
| 0 | 0 | 25 | 26 | 49 | 60 |
| 0 | 0 | 57 | 10 | 26 | 68 |
| 0 | 0 | 106 | 56 | 88 | 87 |
G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,112,0,0,0,0,1,79,0,0,0,0,0,0,0,112,0,0,0,0,1,79],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,1,4,57,0,0,47,105,80,109,0,0,105,112,0,0,0,0,66,8,0,0],[5,7,0,0,0,0,77,108,0,0,0,0,0,0,87,68,57,103,0,0,45,26,10,56,0,0,53,70,26,45,0,0,43,60,68,87],[5,76,0,0,0,0,77,108,0,0,0,0,0,0,87,25,57,106,0,0,45,26,10,56,0,0,53,49,26,88,0,0,43,60,68,87] >;
C7⋊(C8⋊D4) in GAP, Magma, Sage, TeX
C_7\rtimes (C_8\rtimes D_4)
% in TeX
G:=Group("C7:(C8:D4)"); // GroupNames label
G:=SmallGroup(448,346);
// by ID
G=gap.SmallGroup(448,346);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,120,1094,135,184,570,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^8=c^4=d^2=1,b*a*b^-1=d*a*d=a^-1,a*c=c*a,c*b*c^-1=b^-1,d*b*d=b^3,d*c*d=c^-1>;
// generators/relations