direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C8⋊D4, C56⋊19D4, C8⋊1(C7×D4), C2.D8⋊12C14, C4.60(D4×C14), C22⋊Q8⋊4C14, (C2×SD16)⋊1C14, C28.467(C2×D4), C4⋊D4.4C14, (C2×C28).327D4, D4⋊C4⋊17C14, C23.15(C7×D4), Q8⋊C4⋊17C14, (C14×SD16)⋊12C2, (C2×M4(2))⋊1C14, C22.92(D4×C14), (C22×C14).33D4, C28.265(C4○D4), (C14×M4(2))⋊11C2, (C2×C56).332C22, (C2×C28).927C23, C14.151(C4⋊D4), C14.137(C8⋊C22), (D4×C14).191C22, (Q8×C14).165C22, C14.137(C8.C22), (C22×C28).425C22, C4⋊C4.8(C2×C14), (C7×C2.D8)⋊27C2, C4.10(C7×C4○D4), (C2×C4).32(C7×D4), (C2×C8).21(C2×C14), C2.20(C7×C4⋊D4), C2.12(C7×C8⋊C22), (C7×C22⋊Q8)⋊31C2, (C2×Q8).9(C2×C14), (C7×D4⋊C4)⋊40C2, (C7×Q8⋊C4)⋊40C2, (C2×D4).14(C2×C14), (C2×C14).648(C2×D4), (C7×C4⋊D4).14C2, C2.12(C7×C8.C22), (C7×C4⋊C4).230C22, (C22×C4).43(C2×C14), (C2×C4).102(C22×C14), SmallGroup(448,876)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C8⋊D4
G = < a,b,c,d | a7=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=b3, dcd=c-1 >
Subgroups: 234 in 120 conjugacy classes, 54 normal (50 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C14, C14, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C28, C28, C2×C14, C2×C14, D4⋊C4, Q8⋊C4, C2.D8, C4⋊D4, C22⋊Q8, C2×M4(2), C2×SD16, C56, C56, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C22×C14, C8⋊D4, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C2×C56, C7×M4(2), C7×SD16, C22×C28, D4×C14, D4×C14, Q8×C14, C7×D4⋊C4, C7×Q8⋊C4, C7×C2.D8, C7×C4⋊D4, C7×C22⋊Q8, C14×M4(2), C14×SD16, C7×C8⋊D4
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C4○D4, C2×C14, C4⋊D4, C8⋊C22, C8.C22, C7×D4, C22×C14, C8⋊D4, D4×C14, C7×C4○D4, C7×C4⋊D4, C7×C8⋊C22, C7×C8.C22, C7×C8⋊D4
►On 224 points
Generators in S224
(1 113 111 53 103 45 95)(2 114 112 54 104 46 96)(3 115 105 55 97 47 89)(4 116 106 56 98 48 90)(5 117 107 49 99 41 91)(6 118 108 50 100 42 92)(7 119 109 51 101 43 93)(8 120 110 52 102 44 94)(9 84 35 29 75 17 67)(10 85 36 30 76 18 68)(11 86 37 31 77 19 69)(12 87 38 32 78 20 70)(13 88 39 25 79 21 71)(14 81 40 26 80 22 72)(15 82 33 27 73 23 65)(16 83 34 28 74 24 66)(57 217 163 209 155 201 147)(58 218 164 210 156 202 148)(59 219 165 211 157 203 149)(60 220 166 212 158 204 150)(61 221 167 213 159 205 151)(62 222 168 214 160 206 152)(63 223 161 215 153 207 145)(64 224 162 216 154 208 146)(121 175 199 137 191 129 183)(122 176 200 138 192 130 184)(123 169 193 139 185 131 177)(124 170 194 140 186 132 178)(125 171 195 141 187 133 179)(126 172 196 142 188 134 180)(127 173 197 143 189 135 181)(128 174 198 144 190 136 182)
(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)
(1 170 11 151)(2 169 12 150)(3 176 13 149)(4 175 14 148)(5 174 15 147)(6 173 16 146)(7 172 9 145)(8 171 10 152)(17 153 43 180)(18 160 44 179)(19 159 45 178)(20 158 46 177)(21 157 47 184)(22 156 48 183)(23 155 41 182)(24 154 42 181)(25 165 55 192)(26 164 56 191)(27 163 49 190)(28 162 50 189)(29 161 51 188)(30 168 52 187)(31 167 53 186)(32 166 54 185)(33 217 107 144)(34 224 108 143)(35 223 109 142)(36 222 110 141)(37 221 111 140)(38 220 112 139)(39 219 105 138)(40 218 106 137)(57 117 198 82)(58 116 199 81)(59 115 200 88)(60 114 193 87)(61 113 194 86)(62 120 195 85)(63 119 196 84)(64 118 197 83)(65 201 91 128)(66 208 92 127)(67 207 93 126)(68 206 94 125)(69 205 95 124)(70 204 96 123)(71 203 89 122)(72 202 90 121)(73 209 99 136)(74 216 100 135)(75 215 101 134)(76 214 102 133)(77 213 103 132)(78 212 104 131)(79 211 97 130)(80 210 98 129)
(1 151)(2 146)(3 149)(4 152)(5 147)(6 150)(7 145)(8 148)(9 172)(10 175)(11 170)(12 173)(13 176)(14 171)(15 174)(16 169)(17 180)(18 183)(19 178)(20 181)(21 184)(22 179)(23 182)(24 177)(25 192)(26 187)(27 190)(28 185)(29 188)(30 191)(31 186)(32 189)(33 144)(34 139)(35 142)(36 137)(37 140)(38 143)(39 138)(40 141)(41 155)(42 158)(43 153)(44 156)(45 159)(46 154)(47 157)(48 160)(49 163)(50 166)(51 161)(52 164)(53 167)(54 162)(55 165)(56 168)(57 117)(58 120)(59 115)(60 118)(61 113)(62 116)(63 119)(64 114)(65 128)(66 123)(67 126)(68 121)(69 124)(70 127)(71 122)(72 125)(73 136)(74 131)(75 134)(76 129)(77 132)(78 135)(79 130)(80 133)(81 195)(82 198)(83 193)(84 196)(85 199)(86 194)(87 197)(88 200)(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)| (1,113,111,53,103,45,95)(2,114,112,54,104,46,96)(3,115,105,55,97,47,89)(4,116,106,56,98,48,90)(5,117,107,49,99,41,91)(6,118,108,50,100,42,92)(7,119,109,51,101,43,93)(8,120,110,52,102,44,94)(9,84,35,29,75,17,67)(10,85,36,30,76,18,68)(11,86,37,31,77,19,69)(12,87,38,32,78,20,70)(13,88,39,25,79,21,71)(14,81,40,26,80,22,72)(15,82,33,27,73,23,65)(16,83,34,28,74,24,66)(57,217,163,209,155,201,147)(58,218,164,210,156,202,148)(59,219,165,211,157,203,149)(60,220,166,212,158,204,150)(61,221,167,213,159,205,151)(62,222,168,214,160,206,152)(63,223,161,215,153,207,145)(64,224,162,216,154,208,146)(121,175,199,137,191,129,183)(122,176,200,138,192,130,184)(123,169,193,139,185,131,177)(124,170,194,140,186,132,178)(125,171,195,141,187,133,179)(126,172,196,142,188,134,180)(127,173,197,143,189,135,181)(128,174,198,144,190,136,182), (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), (1,170,11,151)(2,169,12,150)(3,176,13,149)(4,175,14,148)(5,174,15,147)(6,173,16,146)(7,172,9,145)(8,171,10,152)(17,153,43,180)(18,160,44,179)(19,159,45,178)(20,158,46,177)(21,157,47,184)(22,156,48,183)(23,155,41,182)(24,154,42,181)(25,165,55,192)(26,164,56,191)(27,163,49,190)(28,162,50,189)(29,161,51,188)(30,168,52,187)(31,167,53,186)(32,166,54,185)(33,217,107,144)(34,224,108,143)(35,223,109,142)(36,222,110,141)(37,221,111,140)(38,220,112,139)(39,219,105,138)(40,218,106,137)(57,117,198,82)(58,116,199,81)(59,115,200,88)(60,114,193,87)(61,113,194,86)(62,120,195,85)(63,119,196,84)(64,118,197,83)(65,201,91,128)(66,208,92,127)(67,207,93,126)(68,206,94,125)(69,205,95,124)(70,204,96,123)(71,203,89,122)(72,202,90,121)(73,209,99,136)(74,216,100,135)(75,215,101,134)(76,214,102,133)(77,213,103,132)(78,212,104,131)(79,211,97,130)(80,210,98,129), (1,151)(2,146)(3,149)(4,152)(5,147)(6,150)(7,145)(8,148)(9,172)(10,175)(11,170)(12,173)(13,176)(14,171)(15,174)(16,169)(17,180)(18,183)(19,178)(20,181)(21,184)(22,179)(23,182)(24,177)(25,192)(26,187)(27,190)(28,185)(29,188)(30,191)(31,186)(32,189)(33,144)(34,139)(35,142)(36,137)(37,140)(38,143)(39,138)(40,141)(41,155)(42,158)(43,153)(44,156)(45,159)(46,154)(47,157)(48,160)(49,163)(50,166)(51,161)(52,164)(53,167)(54,162)(55,165)(56,168)(57,117)(58,120)(59,115)(60,118)(61,113)(62,116)(63,119)(64,114)(65,128)(66,123)(67,126)(68,121)(69,124)(70,127)(71,122)(72,125)(73,136)(74,131)(75,134)(76,129)(77,132)(78,135)(79,130)(80,133)(81,195)(82,198)(83,193)(84,196)(85,199)(86,194)(87,197)(88,200)(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( (1,113,111,53,103,45,95)(2,114,112,54,104,46,96)(3,115,105,55,97,47,89)(4,116,106,56,98,48,90)(5,117,107,49,99,41,91)(6,118,108,50,100,42,92)(7,119,109,51,101,43,93)(8,120,110,52,102,44,94)(9,84,35,29,75,17,67)(10,85,36,30,76,18,68)(11,86,37,31,77,19,69)(12,87,38,32,78,20,70)(13,88,39,25,79,21,71)(14,81,40,26,80,22,72)(15,82,33,27,73,23,65)(16,83,34,28,74,24,66)(57,217,163,209,155,201,147)(58,218,164,210,156,202,148)(59,219,165,211,157,203,149)(60,220,166,212,158,204,150)(61,221,167,213,159,205,151)(62,222,168,214,160,206,152)(63,223,161,215,153,207,145)(64,224,162,216,154,208,146)(121,175,199,137,191,129,183)(122,176,200,138,192,130,184)(123,169,193,139,185,131,177)(124,170,194,140,186,132,178)(125,171,195,141,187,133,179)(126,172,196,142,188,134,180)(127,173,197,143,189,135,181)(128,174,198,144,190,136,182), (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), (1,170,11,151)(2,169,12,150)(3,176,13,149)(4,175,14,148)(5,174,15,147)(6,173,16,146)(7,172,9,145)(8,171,10,152)(17,153,43,180)(18,160,44,179)(19,159,45,178)(20,158,46,177)(21,157,47,184)(22,156,48,183)(23,155,41,182)(24,154,42,181)(25,165,55,192)(26,164,56,191)(27,163,49,190)(28,162,50,189)(29,161,51,188)(30,168,52,187)(31,167,53,186)(32,166,54,185)(33,217,107,144)(34,224,108,143)(35,223,109,142)(36,222,110,141)(37,221,111,140)(38,220,112,139)(39,219,105,138)(40,218,106,137)(57,117,198,82)(58,116,199,81)(59,115,200,88)(60,114,193,87)(61,113,194,86)(62,120,195,85)(63,119,196,84)(64,118,197,83)(65,201,91,128)(66,208,92,127)(67,207,93,126)(68,206,94,125)(69,205,95,124)(70,204,96,123)(71,203,89,122)(72,202,90,121)(73,209,99,136)(74,216,100,135)(75,215,101,134)(76,214,102,133)(77,213,103,132)(78,212,104,131)(79,211,97,130)(80,210,98,129), (1,151)(2,146)(3,149)(4,152)(5,147)(6,150)(7,145)(8,148)(9,172)(10,175)(11,170)(12,173)(13,176)(14,171)(15,174)(16,169)(17,180)(18,183)(19,178)(20,181)(21,184)(22,179)(23,182)(24,177)(25,192)(26,187)(27,190)(28,185)(29,188)(30,191)(31,186)(32,189)(33,144)(34,139)(35,142)(36,137)(37,140)(38,143)(39,138)(40,141)(41,155)(42,158)(43,153)(44,156)(45,159)(46,154)(47,157)(48,160)(49,163)(50,166)(51,161)(52,164)(53,167)(54,162)(55,165)(56,168)(57,117)(58,120)(59,115)(60,118)(61,113)(62,116)(63,119)(64,114)(65,128)(66,123)(67,126)(68,121)(69,124)(70,127)(71,122)(72,125)(73,136)(74,131)(75,134)(76,129)(77,132)(78,135)(79,130)(80,133)(81,195)(82,198)(83,193)(84,196)(85,199)(86,194)(87,197)(88,200)(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([[(1,113,111,53,103,45,95),(2,114,112,54,104,46,96),(3,115,105,55,97,47,89),(4,116,106,56,98,48,90),(5,117,107,49,99,41,91),(6,118,108,50,100,42,92),(7,119,109,51,101,43,93),(8,120,110,52,102,44,94),(9,84,35,29,75,17,67),(10,85,36,30,76,18,68),(11,86,37,31,77,19,69),(12,87,38,32,78,20,70),(13,88,39,25,79,21,71),(14,81,40,26,80,22,72),(15,82,33,27,73,23,65),(16,83,34,28,74,24,66),(57,217,163,209,155,201,147),(58,218,164,210,156,202,148),(59,219,165,211,157,203,149),(60,220,166,212,158,204,150),(61,221,167,213,159,205,151),(62,222,168,214,160,206,152),(63,223,161,215,153,207,145),(64,224,162,216,154,208,146),(121,175,199,137,191,129,183),(122,176,200,138,192,130,184),(123,169,193,139,185,131,177),(124,170,194,140,186,132,178),(125,171,195,141,187,133,179),(126,172,196,142,188,134,180),(127,173,197,143,189,135,181),(128,174,198,144,190,136,182)], [(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)], [(1,170,11,151),(2,169,12,150),(3,176,13,149),(4,175,14,148),(5,174,15,147),(6,173,16,146),(7,172,9,145),(8,171,10,152),(17,153,43,180),(18,160,44,179),(19,159,45,178),(20,158,46,177),(21,157,47,184),(22,156,48,183),(23,155,41,182),(24,154,42,181),(25,165,55,192),(26,164,56,191),(27,163,49,190),(28,162,50,189),(29,161,51,188),(30,168,52,187),(31,167,53,186),(32,166,54,185),(33,217,107,144),(34,224,108,143),(35,223,109,142),(36,222,110,141),(37,221,111,140),(38,220,112,139),(39,219,105,138),(40,218,106,137),(57,117,198,82),(58,116,199,81),(59,115,200,88),(60,114,193,87),(61,113,194,86),(62,120,195,85),(63,119,196,84),(64,118,197,83),(65,201,91,128),(66,208,92,127),(67,207,93,126),(68,206,94,125),(69,205,95,124),(70,204,96,123),(71,203,89,122),(72,202,90,121),(73,209,99,136),(74,216,100,135),(75,215,101,134),(76,214,102,133),(77,213,103,132),(78,212,104,131),(79,211,97,130),(80,210,98,129)], [(1,151),(2,146),(3,149),(4,152),(5,147),(6,150),(7,145),(8,148),(9,172),(10,175),(11,170),(12,173),(13,176),(14,171),(15,174),(16,169),(17,180),(18,183),(19,178),(20,181),(21,184),(22,179),(23,182),(24,177),(25,192),(26,187),(27,190),(28,185),(29,188),(30,191),(31,186),(32,189),(33,144),(34,139),(35,142),(36,137),(37,140),(38,143),(39,138),(40,141),(41,155),(42,158),(43,153),(44,156),(45,159),(46,154),(47,157),(48,160),(49,163),(50,166),(51,161),(52,164),(53,167),(54,162),(55,165),(56,168),(57,117),(58,120),(59,115),(60,118),(61,113),(62,116),(63,119),(64,114),(65,128),(66,123),(67,126),(68,121),(69,124),(70,127),(71,122),(72,125),(73,136),(74,131),(75,134),(76,129),(77,132),(78,135),(79,130),(80,133),(81,195),(82,198),(83,193),(84,196),(85,199),(86,194),(87,197),(88,200),(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)]])
112 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 7A | ··· | 7F | 8A | 8B | 8C | 8D | 14A | ··· | 14R | 14S | ··· | 14X | 14Y | ··· | 14AD | 28A | ··· | 28L | 28M | ··· | 28R | 28S | ··· | 28AJ | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 2 | 2 | 4 | 8 | 8 | 8 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 8 | ··· | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
112 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 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | C14 | C14 | D4 | D4 | D4 | C4○D4 | C7×D4 | C7×D4 | C7×D4 | C7×C4○D4 | C8⋊C22 | C8.C22 | C7×C8⋊C22 | C7×C8.C22 |
| kernel | C7×C8⋊D4 | C7×D4⋊C4 | C7×Q8⋊C4 | C7×C2.D8 | C7×C4⋊D4 | C7×C22⋊Q8 | C14×M4(2) | C14×SD16 | C8⋊D4 | D4⋊C4 | Q8⋊C4 | C2.D8 | C4⋊D4 | C22⋊Q8 | C2×M4(2) | C2×SD16 | C56 | C2×C28 | C22×C14 | C28 | C8 | C2×C4 | C23 | C4 | C14 | C14 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 2 | 1 | 1 | 2 | 12 | 6 | 6 | 12 | 1 | 1 | 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 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 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 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 112 | 112 | 0 | 0 |
| 18 | 111 | 0 | 0 | 0 | 0 |
| 106 | 95 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 99 |
| 0 | 0 | 0 | 0 | 96 | 103 |
| 0 | 0 | 10 | 99 | 0 | 0 |
| 0 | 0 | 96 | 103 | 0 | 0 |
| 18 | 111 | 0 | 0 | 0 | 0 |
| 105 | 95 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 99 |
| 0 | 0 | 0 | 0 | 96 | 103 |
| 0 | 0 | 103 | 14 | 0 | 0 |
| 0 | 0 | 17 | 10 | 0 | 0 |
G:=sub<GL(6,GF(113))| [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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,112,0,0,0,0,2,112,0,0,112,0,0,0,0,0,0,112,0,0],[18,106,0,0,0,0,111,95,0,0,0,0,0,0,0,0,10,96,0,0,0,0,99,103,0,0,10,96,0,0,0,0,99,103,0,0],[18,105,0,0,0,0,111,95,0,0,0,0,0,0,0,0,103,17,0,0,0,0,14,10,0,0,10,96,0,0,0,0,99,103,0,0] >;
C7×C8⋊D4 in GAP, Magma, Sage, TeX
C_7\times C_8\rtimes D_4
% in TeX
G:=Group("C7xC8:D4"); // GroupNames label
G:=SmallGroup(448,876);
// by ID
G=gap.SmallGroup(448,876);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,813,400,2438,2403,9804,172]);
// 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=b^-1,d*b*d=b^3,d*c*d=c^-1>;
// generators/relations