direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8×C7⋊D4, C56⋊31D4, C7⋊5(C8×D4), D14⋊4(C2×C8), C22⋊2(C8×D7), (C22×C8)⋊2D7, D14⋊C8⋊42C2, Dic7⋊2(C2×C8), C14.77(C4×D4), Dic7⋊C8⋊43C2, D14⋊C4.20C4, (C22×C56)⋊15C2, (C8×Dic7)⋊26C2, (C2×C8).345D14, C28.435(C2×D4), C23.33(C4×D7), C14.17(C8○D4), Dic7⋊C4.20C4, C14.20(C22×C8), C23.D7.15C4, C28.251(C4○D4), C4.135(C4○D28), C28.55D4⋊32C2, (C2×C28).859C23, (C2×C56).353C22, (C22×C4).401D14, C2.5(D28.2C4), (C22×C28).560C22, (C4×Dic7).282C22, (D7×C2×C8)⋊24C2, C2.20(D7×C2×C8), (C2×C14)⋊5(C2×C8), C2.3(C4×C7⋊D4), (C2×C4).93(C4×D7), C22.61(C2×C4×D7), (C4×C7⋊D4).19C2, (C2×C7⋊D4).16C4, C4.125(C2×C7⋊D4), (C2×C28).209(C2×C4), (C2×C7⋊C8).320C22, (C2×C4×D7).283C22, (C22×C14).94(C2×C4), (C2×Dic7).66(C2×C4), (C22×D7).43(C2×C4), (C2×C4).801(C22×D7), (C2×C14).129(C22×C4), SmallGroup(448,643)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8×C7⋊D4
G = < a,b,c,d | a8=b7=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 484 in 134 conjugacy classes, 63 normal (47 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×C14, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C22×C8, C7⋊C8, C56, C56, C4×D7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, C8×D4, C8×D7, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, D14⋊C4, C23.D7, C2×C56, C2×C56, C2×C4×D7, C2×C7⋊D4, C22×C28, C8×Dic7, Dic7⋊C8, D14⋊C8, C28.55D4, D7×C2×C8, C4×C7⋊D4, C22×C56, C8×C7⋊D4
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, D7, C2×C8, C22×C4, C2×D4, C4○D4, D14, C4×D4, C22×C8, C8○D4, C4×D7, C7⋊D4, C22×D7, C8×D4, C8×D7, C2×C4×D7, C4○D28, C2×C7⋊D4, D7×C2×C8, D28.2C4, C4×C7⋊D4, C8×C7⋊D4
►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)
(1 51 18 120 179 103 42)(2 52 19 113 180 104 43)(3 53 20 114 181 97 44)(4 54 21 115 182 98 45)(5 55 22 116 183 99 46)(6 56 23 117 184 100 47)(7 49 24 118 177 101 48)(8 50 17 119 178 102 41)(9 75 171 146 72 108 124)(10 76 172 147 65 109 125)(11 77 173 148 66 110 126)(12 78 174 149 67 111 127)(13 79 175 150 68 112 128)(14 80 176 151 69 105 121)(15 73 169 152 70 106 122)(16 74 170 145 71 107 123)(25 92 210 218 83 157 185)(26 93 211 219 84 158 186)(27 94 212 220 85 159 187)(28 95 213 221 86 160 188)(29 96 214 222 87 153 189)(30 89 215 223 88 154 190)(31 90 216 224 81 155 191)(32 91 209 217 82 156 192)(33 206 140 163 129 199 58)(34 207 141 164 130 200 59)(35 208 142 165 131 193 60)(36 201 143 166 132 194 61)(37 202 144 167 133 195 62)(38 203 137 168 134 196 63)(39 204 138 161 135 197 64)(40 205 139 162 136 198 57)
(1 165 69 209)(2 166 70 210)(3 167 71 211)(4 168 72 212)(5 161 65 213)(6 162 66 214)(7 163 67 215)(8 164 68 216)(9 187 115 38)(10 188 116 39)(11 189 117 40)(12 190 118 33)(13 191 119 34)(14 192 120 35)(15 185 113 36)(16 186 114 37)(17 207 128 31)(18 208 121 32)(19 201 122 25)(20 202 123 26)(21 203 124 27)(22 204 125 28)(23 205 126 29)(24 206 127 30)(41 130 150 224)(42 131 151 217)(43 132 152 218)(44 133 145 219)(45 134 146 220)(46 135 147 221)(47 136 148 222)(48 129 149 223)(49 140 111 89)(50 141 112 90)(51 142 105 91)(52 143 106 92)(53 144 107 93)(54 137 108 94)(55 138 109 95)(56 139 110 96)(57 77 153 184)(58 78 154 177)(59 79 155 178)(60 80 156 179)(61 73 157 180)(62 74 158 181)(63 75 159 182)(64 76 160 183)(81 102 200 175)(82 103 193 176)(83 104 194 169)(84 97 195 170)(85 98 196 171)(86 99 197 172)(87 100 198 173)(88 101 199 174)
(1 5)(2 6)(3 7)(4 8)(9 79)(10 80)(11 73)(12 74)(13 75)(14 76)(15 77)(16 78)(17 98)(18 99)(19 100)(20 101)(21 102)(22 103)(23 104)(24 97)(25 198)(26 199)(27 200)(28 193)(29 194)(30 195)(31 196)(32 197)(33 158)(34 159)(35 160)(36 153)(37 154)(38 155)(39 156)(40 157)(41 54)(42 55)(43 56)(44 49)(45 50)(46 51)(47 52)(48 53)(57 185)(58 186)(59 187)(60 188)(61 189)(62 190)(63 191)(64 192)(65 69)(66 70)(67 71)(68 72)(81 203)(82 204)(83 205)(84 206)(85 207)(86 208)(87 201)(88 202)(89 133)(90 134)(91 135)(92 136)(93 129)(94 130)(95 131)(96 132)(105 147)(106 148)(107 149)(108 150)(109 151)(110 152)(111 145)(112 146)(113 184)(114 177)(115 178)(116 179)(117 180)(118 181)(119 182)(120 183)(121 172)(122 173)(123 174)(124 175)(125 176)(126 169)(127 170)(128 171)(137 224)(138 217)(139 218)(140 219)(141 220)(142 221)(143 222)(144 223)(161 209)(162 210)(163 211)(164 212)(165 213)(166 214)(167 215)(168 216)
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), (1,51,18,120,179,103,42)(2,52,19,113,180,104,43)(3,53,20,114,181,97,44)(4,54,21,115,182,98,45)(5,55,22,116,183,99,46)(6,56,23,117,184,100,47)(7,49,24,118,177,101,48)(8,50,17,119,178,102,41)(9,75,171,146,72,108,124)(10,76,172,147,65,109,125)(11,77,173,148,66,110,126)(12,78,174,149,67,111,127)(13,79,175,150,68,112,128)(14,80,176,151,69,105,121)(15,73,169,152,70,106,122)(16,74,170,145,71,107,123)(25,92,210,218,83,157,185)(26,93,211,219,84,158,186)(27,94,212,220,85,159,187)(28,95,213,221,86,160,188)(29,96,214,222,87,153,189)(30,89,215,223,88,154,190)(31,90,216,224,81,155,191)(32,91,209,217,82,156,192)(33,206,140,163,129,199,58)(34,207,141,164,130,200,59)(35,208,142,165,131,193,60)(36,201,143,166,132,194,61)(37,202,144,167,133,195,62)(38,203,137,168,134,196,63)(39,204,138,161,135,197,64)(40,205,139,162,136,198,57), (1,165,69,209)(2,166,70,210)(3,167,71,211)(4,168,72,212)(5,161,65,213)(6,162,66,214)(7,163,67,215)(8,164,68,216)(9,187,115,38)(10,188,116,39)(11,189,117,40)(12,190,118,33)(13,191,119,34)(14,192,120,35)(15,185,113,36)(16,186,114,37)(17,207,128,31)(18,208,121,32)(19,201,122,25)(20,202,123,26)(21,203,124,27)(22,204,125,28)(23,205,126,29)(24,206,127,30)(41,130,150,224)(42,131,151,217)(43,132,152,218)(44,133,145,219)(45,134,146,220)(46,135,147,221)(47,136,148,222)(48,129,149,223)(49,140,111,89)(50,141,112,90)(51,142,105,91)(52,143,106,92)(53,144,107,93)(54,137,108,94)(55,138,109,95)(56,139,110,96)(57,77,153,184)(58,78,154,177)(59,79,155,178)(60,80,156,179)(61,73,157,180)(62,74,158,181)(63,75,159,182)(64,76,160,183)(81,102,200,175)(82,103,193,176)(83,104,194,169)(84,97,195,170)(85,98,196,171)(86,99,197,172)(87,100,198,173)(88,101,199,174), (1,5)(2,6)(3,7)(4,8)(9,79)(10,80)(11,73)(12,74)(13,75)(14,76)(15,77)(16,78)(17,98)(18,99)(19,100)(20,101)(21,102)(22,103)(23,104)(24,97)(25,198)(26,199)(27,200)(28,193)(29,194)(30,195)(31,196)(32,197)(33,158)(34,159)(35,160)(36,153)(37,154)(38,155)(39,156)(40,157)(41,54)(42,55)(43,56)(44,49)(45,50)(46,51)(47,52)(48,53)(57,185)(58,186)(59,187)(60,188)(61,189)(62,190)(63,191)(64,192)(65,69)(66,70)(67,71)(68,72)(81,203)(82,204)(83,205)(84,206)(85,207)(86,208)(87,201)(88,202)(89,133)(90,134)(91,135)(92,136)(93,129)(94,130)(95,131)(96,132)(105,147)(106,148)(107,149)(108,150)(109,151)(110,152)(111,145)(112,146)(113,184)(114,177)(115,178)(116,179)(117,180)(118,181)(119,182)(120,183)(121,172)(122,173)(123,174)(124,175)(125,176)(126,169)(127,170)(128,171)(137,224)(138,217)(139,218)(140,219)(141,220)(142,221)(143,222)(144,223)(161,209)(162,210)(163,211)(164,212)(165,213)(166,214)(167,215)(168,216)>;
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), (1,51,18,120,179,103,42)(2,52,19,113,180,104,43)(3,53,20,114,181,97,44)(4,54,21,115,182,98,45)(5,55,22,116,183,99,46)(6,56,23,117,184,100,47)(7,49,24,118,177,101,48)(8,50,17,119,178,102,41)(9,75,171,146,72,108,124)(10,76,172,147,65,109,125)(11,77,173,148,66,110,126)(12,78,174,149,67,111,127)(13,79,175,150,68,112,128)(14,80,176,151,69,105,121)(15,73,169,152,70,106,122)(16,74,170,145,71,107,123)(25,92,210,218,83,157,185)(26,93,211,219,84,158,186)(27,94,212,220,85,159,187)(28,95,213,221,86,160,188)(29,96,214,222,87,153,189)(30,89,215,223,88,154,190)(31,90,216,224,81,155,191)(32,91,209,217,82,156,192)(33,206,140,163,129,199,58)(34,207,141,164,130,200,59)(35,208,142,165,131,193,60)(36,201,143,166,132,194,61)(37,202,144,167,133,195,62)(38,203,137,168,134,196,63)(39,204,138,161,135,197,64)(40,205,139,162,136,198,57), (1,165,69,209)(2,166,70,210)(3,167,71,211)(4,168,72,212)(5,161,65,213)(6,162,66,214)(7,163,67,215)(8,164,68,216)(9,187,115,38)(10,188,116,39)(11,189,117,40)(12,190,118,33)(13,191,119,34)(14,192,120,35)(15,185,113,36)(16,186,114,37)(17,207,128,31)(18,208,121,32)(19,201,122,25)(20,202,123,26)(21,203,124,27)(22,204,125,28)(23,205,126,29)(24,206,127,30)(41,130,150,224)(42,131,151,217)(43,132,152,218)(44,133,145,219)(45,134,146,220)(46,135,147,221)(47,136,148,222)(48,129,149,223)(49,140,111,89)(50,141,112,90)(51,142,105,91)(52,143,106,92)(53,144,107,93)(54,137,108,94)(55,138,109,95)(56,139,110,96)(57,77,153,184)(58,78,154,177)(59,79,155,178)(60,80,156,179)(61,73,157,180)(62,74,158,181)(63,75,159,182)(64,76,160,183)(81,102,200,175)(82,103,193,176)(83,104,194,169)(84,97,195,170)(85,98,196,171)(86,99,197,172)(87,100,198,173)(88,101,199,174), (1,5)(2,6)(3,7)(4,8)(9,79)(10,80)(11,73)(12,74)(13,75)(14,76)(15,77)(16,78)(17,98)(18,99)(19,100)(20,101)(21,102)(22,103)(23,104)(24,97)(25,198)(26,199)(27,200)(28,193)(29,194)(30,195)(31,196)(32,197)(33,158)(34,159)(35,160)(36,153)(37,154)(38,155)(39,156)(40,157)(41,54)(42,55)(43,56)(44,49)(45,50)(46,51)(47,52)(48,53)(57,185)(58,186)(59,187)(60,188)(61,189)(62,190)(63,191)(64,192)(65,69)(66,70)(67,71)(68,72)(81,203)(82,204)(83,205)(84,206)(85,207)(86,208)(87,201)(88,202)(89,133)(90,134)(91,135)(92,136)(93,129)(94,130)(95,131)(96,132)(105,147)(106,148)(107,149)(108,150)(109,151)(110,152)(111,145)(112,146)(113,184)(114,177)(115,178)(116,179)(117,180)(118,181)(119,182)(120,183)(121,172)(122,173)(123,174)(124,175)(125,176)(126,169)(127,170)(128,171)(137,224)(138,217)(139,218)(140,219)(141,220)(142,221)(143,222)(144,223)(161,209)(162,210)(163,211)(164,212)(165,213)(166,214)(167,215)(168,216) );
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)], [(1,51,18,120,179,103,42),(2,52,19,113,180,104,43),(3,53,20,114,181,97,44),(4,54,21,115,182,98,45),(5,55,22,116,183,99,46),(6,56,23,117,184,100,47),(7,49,24,118,177,101,48),(8,50,17,119,178,102,41),(9,75,171,146,72,108,124),(10,76,172,147,65,109,125),(11,77,173,148,66,110,126),(12,78,174,149,67,111,127),(13,79,175,150,68,112,128),(14,80,176,151,69,105,121),(15,73,169,152,70,106,122),(16,74,170,145,71,107,123),(25,92,210,218,83,157,185),(26,93,211,219,84,158,186),(27,94,212,220,85,159,187),(28,95,213,221,86,160,188),(29,96,214,222,87,153,189),(30,89,215,223,88,154,190),(31,90,216,224,81,155,191),(32,91,209,217,82,156,192),(33,206,140,163,129,199,58),(34,207,141,164,130,200,59),(35,208,142,165,131,193,60),(36,201,143,166,132,194,61),(37,202,144,167,133,195,62),(38,203,137,168,134,196,63),(39,204,138,161,135,197,64),(40,205,139,162,136,198,57)], [(1,165,69,209),(2,166,70,210),(3,167,71,211),(4,168,72,212),(5,161,65,213),(6,162,66,214),(7,163,67,215),(8,164,68,216),(9,187,115,38),(10,188,116,39),(11,189,117,40),(12,190,118,33),(13,191,119,34),(14,192,120,35),(15,185,113,36),(16,186,114,37),(17,207,128,31),(18,208,121,32),(19,201,122,25),(20,202,123,26),(21,203,124,27),(22,204,125,28),(23,205,126,29),(24,206,127,30),(41,130,150,224),(42,131,151,217),(43,132,152,218),(44,133,145,219),(45,134,146,220),(46,135,147,221),(47,136,148,222),(48,129,149,223),(49,140,111,89),(50,141,112,90),(51,142,105,91),(52,143,106,92),(53,144,107,93),(54,137,108,94),(55,138,109,95),(56,139,110,96),(57,77,153,184),(58,78,154,177),(59,79,155,178),(60,80,156,179),(61,73,157,180),(62,74,158,181),(63,75,159,182),(64,76,160,183),(81,102,200,175),(82,103,193,176),(83,104,194,169),(84,97,195,170),(85,98,196,171),(86,99,197,172),(87,100,198,173),(88,101,199,174)], [(1,5),(2,6),(3,7),(4,8),(9,79),(10,80),(11,73),(12,74),(13,75),(14,76),(15,77),(16,78),(17,98),(18,99),(19,100),(20,101),(21,102),(22,103),(23,104),(24,97),(25,198),(26,199),(27,200),(28,193),(29,194),(30,195),(31,196),(32,197),(33,158),(34,159),(35,160),(36,153),(37,154),(38,155),(39,156),(40,157),(41,54),(42,55),(43,56),(44,49),(45,50),(46,51),(47,52),(48,53),(57,185),(58,186),(59,187),(60,188),(61,189),(62,190),(63,191),(64,192),(65,69),(66,70),(67,71),(68,72),(81,203),(82,204),(83,205),(84,206),(85,207),(86,208),(87,201),(88,202),(89,133),(90,134),(91,135),(92,136),(93,129),(94,130),(95,131),(96,132),(105,147),(106,148),(107,149),(108,150),(109,151),(110,152),(111,145),(112,146),(113,184),(114,177),(115,178),(116,179),(117,180),(118,181),(119,182),(120,183),(121,172),(122,173),(123,174),(124,175),(125,176),(126,169),(127,170),(128,171),(137,224),(138,217),(139,218),(140,219),(141,220),(142,221),(143,222),(144,223),(161,209),(162,210),(163,211),(164,212),(165,213),(166,214),(167,215),(168,216)]])
136 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 7A | 7B | 7C | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 8M | ··· | 8T | 14A | ··· | 14U | 28A | ··· | 28X | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 14 | 14 | 1 | 1 | 1 | 1 | 2 | 2 | 14 | ··· | 14 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
136 irreducible representations
| dim | 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 | C4 | C4 | C4 | C4 | C8 | D4 | D7 | C4○D4 | D14 | D14 | C8○D4 | C7⋊D4 | C4×D7 | C4×D7 | C4○D28 | C8×D7 | D28.2C4 |
| kernel | C8×C7⋊D4 | C8×Dic7 | Dic7⋊C8 | D14⋊C8 | C28.55D4 | D7×C2×C8 | C4×C7⋊D4 | C22×C56 | Dic7⋊C4 | D14⋊C4 | C23.D7 | C2×C7⋊D4 | C7⋊D4 | C56 | C22×C8 | C28 | C2×C8 | C22×C4 | C14 | C8 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 16 | 2 | 3 | 2 | 6 | 3 | 4 | 12 | 6 | 6 | 12 | 24 | 24 |
Matrix representation of C8×C7⋊D4 ►in GL3(𝔽113) generated by
| 18 | 0 | 0 |
| 0 | 18 | 0 |
| 0 | 0 | 18 |
| 1 | 0 | 0 |
| 0 | 112 | 87 |
| 0 | 1 | 25 |
| 1 | 0 | 0 |
| 0 | 59 | 28 |
| 0 | 29 | 54 |
| 1 | 0 | 0 |
| 0 | 25 | 59 |
| 0 | 112 | 88 |
G:=sub<GL(3,GF(113))| [18,0,0,0,18,0,0,0,18],[1,0,0,0,112,1,0,87,25],[1,0,0,0,59,29,0,28,54],[1,0,0,0,25,112,0,59,88] >;
C8×C7⋊D4 in GAP, Magma, Sage, TeX
C_8\times C_7\rtimes D_4
% in TeX
G:=Group("C8xC7:D4"); // GroupNames label
G:=SmallGroup(448,643);
// by ID
G=gap.SmallGroup(448,643);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,58,136,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^7=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^-1,d*c*d=c^-1>;
// generators/relations