direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C8⋊8D4, C56⋊34D4, C8⋊8(C7×D4), C4.Q8⋊6C14, (C22×C8)⋊9C14, (C2×C14)⋊7SD16, C4.57(D4×C14), C22⋊Q8⋊3C14, D4⋊C4⋊1C14, (C22×C56)⋊23C2, Q8⋊C4⋊1C14, C28.464(C2×D4), C4⋊D4.3C14, (C2×C28).363D4, C2.9(C14×SD16), C22⋊1(C7×SD16), C23.25(C7×D4), (C2×SD16)⋊13C14, (C14×SD16)⋊30C2, C14.89(C2×SD16), C22.89(D4×C14), C28.262(C4○D4), C14.123(C4○D8), (C2×C56).363C22, (C2×C28).924C23, (C22×C14).129D4, C14.148(C4⋊D4), (D4×C14).189C22, (Q8×C14).163C22, (C22×C28).591C22, C4.7(C7×C4○D4), C4⋊C4.5(C2×C14), (C7×C4.Q8)⋊21C2, C2.10(C7×C4○D8), (C2×C4).53(C7×D4), (C7×D4⋊C4)⋊1C2, (C2×C8).93(C2×C14), (C7×Q8⋊C4)⋊1C2, C2.17(C7×C4⋊D4), (C7×C22⋊Q8)⋊30C2, (C2×Q8).7(C2×C14), (C2×D4).12(C2×C14), (C7×C4⋊D4).13C2, (C2×C14).645(C2×D4), (C7×C4⋊C4).227C22, (C2×C4).99(C22×C14), (C22×C4).120(C2×C14), SmallGroup(448,873)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C8⋊8D4
G = < a,b,c,d | a7=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b3, dcd=c-1 >
Subgroups: 234 in 124 conjugacy classes, 58 normal (50 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C14, C14, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C28, C28, C2×C14, C2×C14, C2×C14, D4⋊C4, Q8⋊C4, C4.Q8, C4⋊D4, C22⋊Q8, C22×C8, C2×SD16, C56, C56, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C22×C14, C8⋊8D4, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C2×C56, C2×C56, C7×SD16, C22×C28, D4×C14, D4×C14, Q8×C14, C7×D4⋊C4, C7×Q8⋊C4, C7×C4.Q8, C7×C4⋊D4, C7×C22⋊Q8, C22×C56, C14×SD16, C7×C8⋊8D4
Quotients: C1, C2, C22, C7, D4, C23, C14, SD16, C2×D4, C4○D4, C2×C14, C4⋊D4, C2×SD16, C4○D8, C7×D4, C22×C14, C8⋊8D4, C7×SD16, D4×C14, C7×C4○D4, C7×C4⋊D4, C14×SD16, C7×C4○D8, C7×C8⋊8D4
►On 224 points
Generators in S224
(1 14 71 87 31 79 23)(2 15 72 88 32 80 24)(3 16 65 81 25 73 17)(4 9 66 82 26 74 18)(5 10 67 83 27 75 19)(6 11 68 84 28 76 20)(7 12 69 85 29 77 21)(8 13 70 86 30 78 22)(33 117 105 49 97 41 89)(34 118 106 50 98 42 90)(35 119 107 51 99 43 91)(36 120 108 52 100 44 92)(37 113 109 53 101 45 93)(38 114 110 54 102 46 94)(39 115 111 55 103 47 95)(40 116 112 56 104 48 96)(57 219 167 211 159 203 151)(58 220 168 212 160 204 152)(59 221 161 213 153 205 145)(60 222 162 214 154 206 146)(61 223 163 215 155 207 147)(62 224 164 216 156 208 148)(63 217 165 209 157 201 149)(64 218 166 210 158 202 150)(121 173 200 140 192 132 184)(122 174 193 141 185 133 177)(123 175 194 142 186 134 178)(124 176 195 143 187 135 179)(125 169 196 144 188 136 180)(126 170 197 137 189 129 181)(127 171 198 138 190 130 182)(128 172 199 139 191 131 183)
(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)
(1 145 39 177)(2 148 40 180)(3 151 33 183)(4 146 34 178)(5 149 35 181)(6 152 36 184)(7 147 37 179)(8 150 38 182)(9 60 118 123)(10 63 119 126)(11 58 120 121)(12 61 113 124)(13 64 114 127)(14 59 115 122)(15 62 116 125)(16 57 117 128)(17 203 89 131)(18 206 90 134)(19 201 91 129)(20 204 92 132)(21 207 93 135)(22 202 94 130)(23 205 95 133)(24 208 96 136)(25 211 97 139)(26 214 98 142)(27 209 99 137)(28 212 100 140)(29 215 101 143)(30 210 102 138)(31 213 103 141)(32 216 104 144)(41 191 73 159)(42 186 74 154)(43 189 75 157)(44 192 76 160)(45 187 77 155)(46 190 78 158)(47 185 79 153)(48 188 80 156)(49 199 81 167)(50 194 82 162)(51 197 83 165)(52 200 84 168)(53 195 85 163)(54 198 86 166)(55 193 87 161)(56 196 88 164)(65 219 105 172)(66 222 106 175)(67 217 107 170)(68 220 108 173)(69 223 109 176)(70 218 110 171)(71 221 111 174)(72 224 112 169)
(1 177)(2 180)(3 183)(4 178)(5 181)(6 184)(7 179)(8 182)(9 123)(10 126)(11 121)(12 124)(13 127)(14 122)(15 125)(16 128)(17 131)(18 134)(19 129)(20 132)(21 135)(22 130)(23 133)(24 136)(25 139)(26 142)(27 137)(28 140)(29 143)(30 138)(31 141)(32 144)(33 151)(34 146)(35 149)(36 152)(37 147)(38 150)(39 145)(40 148)(41 159)(42 154)(43 157)(44 160)(45 155)(46 158)(47 153)(48 156)(49 167)(50 162)(51 165)(52 168)(53 163)(54 166)(55 161)(56 164)(57 117)(58 120)(59 115)(60 118)(61 113)(62 116)(63 119)(64 114)(65 172)(66 175)(67 170)(68 173)(69 176)(70 171)(71 174)(72 169)(73 191)(74 186)(75 189)(76 192)(77 187)(78 190)(79 185)(80 188)(81 199)(82 194)(83 197)(84 200)(85 195)(86 198)(87 193)(88 196)(89 203)(90 206)(91 201)(92 204)(93 207)(94 202)(95 205)(96 208)(97 211)(98 214)(99 209)(100 212)(101 215)(102 210)(103 213)(104 216)(105 219)(106 222)(107 217)(108 220)(109 223)(110 218)(111 221)(112 224)
G:=sub<Sym(224)| (1,14,71,87,31,79,23)(2,15,72,88,32,80,24)(3,16,65,81,25,73,17)(4,9,66,82,26,74,18)(5,10,67,83,27,75,19)(6,11,68,84,28,76,20)(7,12,69,85,29,77,21)(8,13,70,86,30,78,22)(33,117,105,49,97,41,89)(34,118,106,50,98,42,90)(35,119,107,51,99,43,91)(36,120,108,52,100,44,92)(37,113,109,53,101,45,93)(38,114,110,54,102,46,94)(39,115,111,55,103,47,95)(40,116,112,56,104,48,96)(57,219,167,211,159,203,151)(58,220,168,212,160,204,152)(59,221,161,213,153,205,145)(60,222,162,214,154,206,146)(61,223,163,215,155,207,147)(62,224,164,216,156,208,148)(63,217,165,209,157,201,149)(64,218,166,210,158,202,150)(121,173,200,140,192,132,184)(122,174,193,141,185,133,177)(123,175,194,142,186,134,178)(124,176,195,143,187,135,179)(125,169,196,144,188,136,180)(126,170,197,137,189,129,181)(127,171,198,138,190,130,182)(128,172,199,139,191,131,183), (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), (1,145,39,177)(2,148,40,180)(3,151,33,183)(4,146,34,178)(5,149,35,181)(6,152,36,184)(7,147,37,179)(8,150,38,182)(9,60,118,123)(10,63,119,126)(11,58,120,121)(12,61,113,124)(13,64,114,127)(14,59,115,122)(15,62,116,125)(16,57,117,128)(17,203,89,131)(18,206,90,134)(19,201,91,129)(20,204,92,132)(21,207,93,135)(22,202,94,130)(23,205,95,133)(24,208,96,136)(25,211,97,139)(26,214,98,142)(27,209,99,137)(28,212,100,140)(29,215,101,143)(30,210,102,138)(31,213,103,141)(32,216,104,144)(41,191,73,159)(42,186,74,154)(43,189,75,157)(44,192,76,160)(45,187,77,155)(46,190,78,158)(47,185,79,153)(48,188,80,156)(49,199,81,167)(50,194,82,162)(51,197,83,165)(52,200,84,168)(53,195,85,163)(54,198,86,166)(55,193,87,161)(56,196,88,164)(65,219,105,172)(66,222,106,175)(67,217,107,170)(68,220,108,173)(69,223,109,176)(70,218,110,171)(71,221,111,174)(72,224,112,169), (1,177)(2,180)(3,183)(4,178)(5,181)(6,184)(7,179)(8,182)(9,123)(10,126)(11,121)(12,124)(13,127)(14,122)(15,125)(16,128)(17,131)(18,134)(19,129)(20,132)(21,135)(22,130)(23,133)(24,136)(25,139)(26,142)(27,137)(28,140)(29,143)(30,138)(31,141)(32,144)(33,151)(34,146)(35,149)(36,152)(37,147)(38,150)(39,145)(40,148)(41,159)(42,154)(43,157)(44,160)(45,155)(46,158)(47,153)(48,156)(49,167)(50,162)(51,165)(52,168)(53,163)(54,166)(55,161)(56,164)(57,117)(58,120)(59,115)(60,118)(61,113)(62,116)(63,119)(64,114)(65,172)(66,175)(67,170)(68,173)(69,176)(70,171)(71,174)(72,169)(73,191)(74,186)(75,189)(76,192)(77,187)(78,190)(79,185)(80,188)(81,199)(82,194)(83,197)(84,200)(85,195)(86,198)(87,193)(88,196)(89,203)(90,206)(91,201)(92,204)(93,207)(94,202)(95,205)(96,208)(97,211)(98,214)(99,209)(100,212)(101,215)(102,210)(103,213)(104,216)(105,219)(106,222)(107,217)(108,220)(109,223)(110,218)(111,221)(112,224)>;
G:=Group( (1,14,71,87,31,79,23)(2,15,72,88,32,80,24)(3,16,65,81,25,73,17)(4,9,66,82,26,74,18)(5,10,67,83,27,75,19)(6,11,68,84,28,76,20)(7,12,69,85,29,77,21)(8,13,70,86,30,78,22)(33,117,105,49,97,41,89)(34,118,106,50,98,42,90)(35,119,107,51,99,43,91)(36,120,108,52,100,44,92)(37,113,109,53,101,45,93)(38,114,110,54,102,46,94)(39,115,111,55,103,47,95)(40,116,112,56,104,48,96)(57,219,167,211,159,203,151)(58,220,168,212,160,204,152)(59,221,161,213,153,205,145)(60,222,162,214,154,206,146)(61,223,163,215,155,207,147)(62,224,164,216,156,208,148)(63,217,165,209,157,201,149)(64,218,166,210,158,202,150)(121,173,200,140,192,132,184)(122,174,193,141,185,133,177)(123,175,194,142,186,134,178)(124,176,195,143,187,135,179)(125,169,196,144,188,136,180)(126,170,197,137,189,129,181)(127,171,198,138,190,130,182)(128,172,199,139,191,131,183), (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), (1,145,39,177)(2,148,40,180)(3,151,33,183)(4,146,34,178)(5,149,35,181)(6,152,36,184)(7,147,37,179)(8,150,38,182)(9,60,118,123)(10,63,119,126)(11,58,120,121)(12,61,113,124)(13,64,114,127)(14,59,115,122)(15,62,116,125)(16,57,117,128)(17,203,89,131)(18,206,90,134)(19,201,91,129)(20,204,92,132)(21,207,93,135)(22,202,94,130)(23,205,95,133)(24,208,96,136)(25,211,97,139)(26,214,98,142)(27,209,99,137)(28,212,100,140)(29,215,101,143)(30,210,102,138)(31,213,103,141)(32,216,104,144)(41,191,73,159)(42,186,74,154)(43,189,75,157)(44,192,76,160)(45,187,77,155)(46,190,78,158)(47,185,79,153)(48,188,80,156)(49,199,81,167)(50,194,82,162)(51,197,83,165)(52,200,84,168)(53,195,85,163)(54,198,86,166)(55,193,87,161)(56,196,88,164)(65,219,105,172)(66,222,106,175)(67,217,107,170)(68,220,108,173)(69,223,109,176)(70,218,110,171)(71,221,111,174)(72,224,112,169), (1,177)(2,180)(3,183)(4,178)(5,181)(6,184)(7,179)(8,182)(9,123)(10,126)(11,121)(12,124)(13,127)(14,122)(15,125)(16,128)(17,131)(18,134)(19,129)(20,132)(21,135)(22,130)(23,133)(24,136)(25,139)(26,142)(27,137)(28,140)(29,143)(30,138)(31,141)(32,144)(33,151)(34,146)(35,149)(36,152)(37,147)(38,150)(39,145)(40,148)(41,159)(42,154)(43,157)(44,160)(45,155)(46,158)(47,153)(48,156)(49,167)(50,162)(51,165)(52,168)(53,163)(54,166)(55,161)(56,164)(57,117)(58,120)(59,115)(60,118)(61,113)(62,116)(63,119)(64,114)(65,172)(66,175)(67,170)(68,173)(69,176)(70,171)(71,174)(72,169)(73,191)(74,186)(75,189)(76,192)(77,187)(78,190)(79,185)(80,188)(81,199)(82,194)(83,197)(84,200)(85,195)(86,198)(87,193)(88,196)(89,203)(90,206)(91,201)(92,204)(93,207)(94,202)(95,205)(96,208)(97,211)(98,214)(99,209)(100,212)(101,215)(102,210)(103,213)(104,216)(105,219)(106,222)(107,217)(108,220)(109,223)(110,218)(111,221)(112,224) );
G=PermutationGroup([[(1,14,71,87,31,79,23),(2,15,72,88,32,80,24),(3,16,65,81,25,73,17),(4,9,66,82,26,74,18),(5,10,67,83,27,75,19),(6,11,68,84,28,76,20),(7,12,69,85,29,77,21),(8,13,70,86,30,78,22),(33,117,105,49,97,41,89),(34,118,106,50,98,42,90),(35,119,107,51,99,43,91),(36,120,108,52,100,44,92),(37,113,109,53,101,45,93),(38,114,110,54,102,46,94),(39,115,111,55,103,47,95),(40,116,112,56,104,48,96),(57,219,167,211,159,203,151),(58,220,168,212,160,204,152),(59,221,161,213,153,205,145),(60,222,162,214,154,206,146),(61,223,163,215,155,207,147),(62,224,164,216,156,208,148),(63,217,165,209,157,201,149),(64,218,166,210,158,202,150),(121,173,200,140,192,132,184),(122,174,193,141,185,133,177),(123,175,194,142,186,134,178),(124,176,195,143,187,135,179),(125,169,196,144,188,136,180),(126,170,197,137,189,129,181),(127,171,198,138,190,130,182),(128,172,199,139,191,131,183)], [(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)], [(1,145,39,177),(2,148,40,180),(3,151,33,183),(4,146,34,178),(5,149,35,181),(6,152,36,184),(7,147,37,179),(8,150,38,182),(9,60,118,123),(10,63,119,126),(11,58,120,121),(12,61,113,124),(13,64,114,127),(14,59,115,122),(15,62,116,125),(16,57,117,128),(17,203,89,131),(18,206,90,134),(19,201,91,129),(20,204,92,132),(21,207,93,135),(22,202,94,130),(23,205,95,133),(24,208,96,136),(25,211,97,139),(26,214,98,142),(27,209,99,137),(28,212,100,140),(29,215,101,143),(30,210,102,138),(31,213,103,141),(32,216,104,144),(41,191,73,159),(42,186,74,154),(43,189,75,157),(44,192,76,160),(45,187,77,155),(46,190,78,158),(47,185,79,153),(48,188,80,156),(49,199,81,167),(50,194,82,162),(51,197,83,165),(52,200,84,168),(53,195,85,163),(54,198,86,166),(55,193,87,161),(56,196,88,164),(65,219,105,172),(66,222,106,175),(67,217,107,170),(68,220,108,173),(69,223,109,176),(70,218,110,171),(71,221,111,174),(72,224,112,169)], [(1,177),(2,180),(3,183),(4,178),(5,181),(6,184),(7,179),(8,182),(9,123),(10,126),(11,121),(12,124),(13,127),(14,122),(15,125),(16,128),(17,131),(18,134),(19,129),(20,132),(21,135),(22,130),(23,133),(24,136),(25,139),(26,142),(27,137),(28,140),(29,143),(30,138),(31,141),(32,144),(33,151),(34,146),(35,149),(36,152),(37,147),(38,150),(39,145),(40,148),(41,159),(42,154),(43,157),(44,160),(45,155),(46,158),(47,153),(48,156),(49,167),(50,162),(51,165),(52,168),(53,163),(54,166),(55,161),(56,164),(57,117),(58,120),(59,115),(60,118),(61,113),(62,116),(63,119),(64,114),(65,172),(66,175),(67,170),(68,173),(69,176),(70,171),(71,174),(72,169),(73,191),(74,186),(75,189),(76,192),(77,187),(78,190),(79,185),(80,188),(81,199),(82,194),(83,197),(84,200),(85,195),(86,198),(87,193),(88,196),(89,203),(90,206),(91,201),(92,204),(93,207),(94,202),(95,205),(96,208),(97,211),(98,214),(99,209),(100,212),(101,215),(102,210),(103,213),(104,216),(105,219),(106,222),(107,217),(108,220),(109,223),(110,218),(111,221),(112,224)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | ··· | 7F | 8A | ··· | 8H | 14A | ··· | 14R | 14S | ··· | 14AD | 14AE | ··· | 14AJ | 28A | ··· | 28X | 28Y | ··· | 28AP | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | C14 | C14 | D4 | D4 | D4 | C4○D4 | SD16 | C4○D8 | C7×D4 | C7×D4 | C7×D4 | C7×C4○D4 | C7×SD16 | C7×C4○D8 |
| kernel | C7×C8⋊8D4 | C7×D4⋊C4 | C7×Q8⋊C4 | C7×C4.Q8 | C7×C4⋊D4 | C7×C22⋊Q8 | C22×C56 | C14×SD16 | C8⋊8D4 | D4⋊C4 | Q8⋊C4 | C4.Q8 | C4⋊D4 | C22⋊Q8 | C22×C8 | C2×SD16 | C56 | C2×C28 | C22×C14 | C28 | C2×C14 | C14 | C8 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 2 | 1 | 1 | 2 | 4 | 4 | 12 | 6 | 6 | 12 | 24 | 24 |
Matrix representation of C7×C8⋊8D4 ►in GL4(𝔽113) generated by
| 49 | 0 | 0 | 0 |
| 0 | 49 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 44 | 0 | 0 | 0 |
| 9 | 95 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 106 | 111 | 0 | 0 |
| 25 | 7 | 0 | 0 |
| 0 | 0 | 54 | 21 |
| 0 | 0 | 1 | 59 |
| 106 | 111 | 0 | 0 |
| 24 | 7 | 0 | 0 |
| 0 | 0 | 59 | 90 |
| 0 | 0 | 112 | 54 |
G:=sub<GL(4,GF(113))| [49,0,0,0,0,49,0,0,0,0,16,0,0,0,0,16],[44,9,0,0,0,95,0,0,0,0,1,0,0,0,0,1],[106,25,0,0,111,7,0,0,0,0,54,1,0,0,21,59],[106,24,0,0,111,7,0,0,0,0,59,112,0,0,90,54] >;
C7×C8⋊8D4 in GAP, Magma, Sage, TeX
C_7\times C_8\rtimes_8D_4
% in TeX
G:=Group("C7xC8:8D4"); // GroupNames label
G:=SmallGroup(448,873);
// by ID
G=gap.SmallGroup(448,873);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,813,400,2438,9804,172]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^8=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^3,d*c*d=c^-1>;
// generators/relations