metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56⋊17D4, D14⋊4M4(2), C7⋊7(C8⋊9D4), D14⋊C8⋊38C2, C8⋊12(C7⋊D4), C56⋊C4⋊28C2, C14.79(C4×D4), Dic7⋊C8⋊40C2, D14⋊C4.16C4, C28.442(C2×D4), (C2×C8).188D14, (C2×M4(2))⋊7D7, C23.19(C4×D7), C14.32(C8○D4), (C14×M4(2))⋊6C2, Dic7⋊C4.16C4, C2.21(D7×M4(2)), C23.D7.12C4, C4.137(C4○D28), C28.253(C4○D4), C28.55D4⋊29C2, C2.17(D28.C4), (C2×C56).234C22, (C2×C28).867C23, (C22×C4).136D14, C14.32(C2×M4(2)), (C22×C28).374C22, (C4×Dic7).189C22, (D7×C2×C8)⋊27C2, (C2×C4).50(C4×D7), C2.24(C4×C7⋊D4), (C2×C7⋊D4).10C4, (C4×C7⋊D4).16C2, C4.133(C2×C7⋊D4), C22.146(C2×C4×D7), (C2×C28).187(C2×C4), (C2×C7⋊C8).323C22, (C2×C4×D7).285C22, (C22×C14).67(C2×C4), (C2×Dic7).34(C2×C4), (C22×D7).44(C2×C4), (C2×C4).809(C22×D7), (C2×C14).137(C22×C4), SmallGroup(448,661)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56⋊D4
G = < a,b,c | a56=b4=c2=1, bab-1=a13, cac=a41, cbc=b-1 >
Subgroups: 484 in 124 conjugacy classes, 53 normal (47 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C8⋊C4, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C2×M4(2), C7⋊C8, C56, C56, C4×D7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×C14, C8⋊9D4, C8×D7, C2×C7⋊C8, C4×Dic7, Dic7⋊C4, D14⋊C4, C23.D7, C2×C56, C7×M4(2), C2×C4×D7, C2×C7⋊D4, C22×C28, Dic7⋊C8, C56⋊C4, D14⋊C8, C28.55D4, D7×C2×C8, C4×C7⋊D4, C14×M4(2), C56⋊D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, M4(2), C22×C4, C2×D4, C4○D4, D14, C4×D4, C2×M4(2), C8○D4, C4×D7, C7⋊D4, C22×D7, C8⋊9D4, C2×C4×D7, C4○D28, C2×C7⋊D4, D7×M4(2), D28.C4, C4×C7⋊D4, C56⋊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)
(1 195 137 57)(2 208 138 70)(3 221 139 83)(4 178 140 96)(5 191 141 109)(6 204 142 66)(7 217 143 79)(8 174 144 92)(9 187 145 105)(10 200 146 62)(11 213 147 75)(12 170 148 88)(13 183 149 101)(14 196 150 58)(15 209 151 71)(16 222 152 84)(17 179 153 97)(18 192 154 110)(19 205 155 67)(20 218 156 80)(21 175 157 93)(22 188 158 106)(23 201 159 63)(24 214 160 76)(25 171 161 89)(26 184 162 102)(27 197 163 59)(28 210 164 72)(29 223 165 85)(30 180 166 98)(31 193 167 111)(32 206 168 68)(33 219 113 81)(34 176 114 94)(35 189 115 107)(36 202 116 64)(37 215 117 77)(38 172 118 90)(39 185 119 103)(40 198 120 60)(41 211 121 73)(42 224 122 86)(43 181 123 99)(44 194 124 112)(45 207 125 69)(46 220 126 82)(47 177 127 95)(48 190 128 108)(49 203 129 65)(50 216 130 78)(51 173 131 91)(52 186 132 104)(53 199 133 61)(54 212 134 74)(55 169 135 87)(56 182 136 100)
(2 42)(3 27)(4 12)(5 53)(6 38)(7 23)(9 49)(10 34)(11 19)(13 45)(14 30)(16 56)(17 41)(18 26)(20 52)(21 37)(24 48)(25 33)(28 44)(31 55)(32 40)(35 51)(39 47)(46 54)(57 195)(58 180)(59 221)(60 206)(61 191)(62 176)(63 217)(64 202)(65 187)(66 172)(67 213)(68 198)(69 183)(70 224)(71 209)(72 194)(73 179)(74 220)(75 205)(76 190)(77 175)(78 216)(79 201)(80 186)(81 171)(82 212)(83 197)(84 182)(85 223)(86 208)(87 193)(88 178)(89 219)(90 204)(91 189)(92 174)(93 215)(94 200)(95 185)(96 170)(97 211)(98 196)(99 181)(100 222)(101 207)(102 192)(103 177)(104 218)(105 203)(106 188)(107 173)(108 214)(109 199)(110 184)(111 169)(112 210)(113 161)(114 146)(115 131)(117 157)(118 142)(119 127)(120 168)(121 153)(122 138)(124 164)(125 149)(126 134)(128 160)(129 145)(132 156)(133 141)(135 167)(136 152)(139 163)(140 148)(143 159)(147 155)(150 166)(154 162)
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), (1,195,137,57)(2,208,138,70)(3,221,139,83)(4,178,140,96)(5,191,141,109)(6,204,142,66)(7,217,143,79)(8,174,144,92)(9,187,145,105)(10,200,146,62)(11,213,147,75)(12,170,148,88)(13,183,149,101)(14,196,150,58)(15,209,151,71)(16,222,152,84)(17,179,153,97)(18,192,154,110)(19,205,155,67)(20,218,156,80)(21,175,157,93)(22,188,158,106)(23,201,159,63)(24,214,160,76)(25,171,161,89)(26,184,162,102)(27,197,163,59)(28,210,164,72)(29,223,165,85)(30,180,166,98)(31,193,167,111)(32,206,168,68)(33,219,113,81)(34,176,114,94)(35,189,115,107)(36,202,116,64)(37,215,117,77)(38,172,118,90)(39,185,119,103)(40,198,120,60)(41,211,121,73)(42,224,122,86)(43,181,123,99)(44,194,124,112)(45,207,125,69)(46,220,126,82)(47,177,127,95)(48,190,128,108)(49,203,129,65)(50,216,130,78)(51,173,131,91)(52,186,132,104)(53,199,133,61)(54,212,134,74)(55,169,135,87)(56,182,136,100), (2,42)(3,27)(4,12)(5,53)(6,38)(7,23)(9,49)(10,34)(11,19)(13,45)(14,30)(16,56)(17,41)(18,26)(20,52)(21,37)(24,48)(25,33)(28,44)(31,55)(32,40)(35,51)(39,47)(46,54)(57,195)(58,180)(59,221)(60,206)(61,191)(62,176)(63,217)(64,202)(65,187)(66,172)(67,213)(68,198)(69,183)(70,224)(71,209)(72,194)(73,179)(74,220)(75,205)(76,190)(77,175)(78,216)(79,201)(80,186)(81,171)(82,212)(83,197)(84,182)(85,223)(86,208)(87,193)(88,178)(89,219)(90,204)(91,189)(92,174)(93,215)(94,200)(95,185)(96,170)(97,211)(98,196)(99,181)(100,222)(101,207)(102,192)(103,177)(104,218)(105,203)(106,188)(107,173)(108,214)(109,199)(110,184)(111,169)(112,210)(113,161)(114,146)(115,131)(117,157)(118,142)(119,127)(120,168)(121,153)(122,138)(124,164)(125,149)(126,134)(128,160)(129,145)(132,156)(133,141)(135,167)(136,152)(139,163)(140,148)(143,159)(147,155)(150,166)(154,162)>;
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), (1,195,137,57)(2,208,138,70)(3,221,139,83)(4,178,140,96)(5,191,141,109)(6,204,142,66)(7,217,143,79)(8,174,144,92)(9,187,145,105)(10,200,146,62)(11,213,147,75)(12,170,148,88)(13,183,149,101)(14,196,150,58)(15,209,151,71)(16,222,152,84)(17,179,153,97)(18,192,154,110)(19,205,155,67)(20,218,156,80)(21,175,157,93)(22,188,158,106)(23,201,159,63)(24,214,160,76)(25,171,161,89)(26,184,162,102)(27,197,163,59)(28,210,164,72)(29,223,165,85)(30,180,166,98)(31,193,167,111)(32,206,168,68)(33,219,113,81)(34,176,114,94)(35,189,115,107)(36,202,116,64)(37,215,117,77)(38,172,118,90)(39,185,119,103)(40,198,120,60)(41,211,121,73)(42,224,122,86)(43,181,123,99)(44,194,124,112)(45,207,125,69)(46,220,126,82)(47,177,127,95)(48,190,128,108)(49,203,129,65)(50,216,130,78)(51,173,131,91)(52,186,132,104)(53,199,133,61)(54,212,134,74)(55,169,135,87)(56,182,136,100), (2,42)(3,27)(4,12)(5,53)(6,38)(7,23)(9,49)(10,34)(11,19)(13,45)(14,30)(16,56)(17,41)(18,26)(20,52)(21,37)(24,48)(25,33)(28,44)(31,55)(32,40)(35,51)(39,47)(46,54)(57,195)(58,180)(59,221)(60,206)(61,191)(62,176)(63,217)(64,202)(65,187)(66,172)(67,213)(68,198)(69,183)(70,224)(71,209)(72,194)(73,179)(74,220)(75,205)(76,190)(77,175)(78,216)(79,201)(80,186)(81,171)(82,212)(83,197)(84,182)(85,223)(86,208)(87,193)(88,178)(89,219)(90,204)(91,189)(92,174)(93,215)(94,200)(95,185)(96,170)(97,211)(98,196)(99,181)(100,222)(101,207)(102,192)(103,177)(104,218)(105,203)(106,188)(107,173)(108,214)(109,199)(110,184)(111,169)(112,210)(113,161)(114,146)(115,131)(117,157)(118,142)(119,127)(120,168)(121,153)(122,138)(124,164)(125,149)(126,134)(128,160)(129,145)(132,156)(133,141)(135,167)(136,152)(139,163)(140,148)(143,159)(147,155)(150,166)(154,162) );
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)], [(1,195,137,57),(2,208,138,70),(3,221,139,83),(4,178,140,96),(5,191,141,109),(6,204,142,66),(7,217,143,79),(8,174,144,92),(9,187,145,105),(10,200,146,62),(11,213,147,75),(12,170,148,88),(13,183,149,101),(14,196,150,58),(15,209,151,71),(16,222,152,84),(17,179,153,97),(18,192,154,110),(19,205,155,67),(20,218,156,80),(21,175,157,93),(22,188,158,106),(23,201,159,63),(24,214,160,76),(25,171,161,89),(26,184,162,102),(27,197,163,59),(28,210,164,72),(29,223,165,85),(30,180,166,98),(31,193,167,111),(32,206,168,68),(33,219,113,81),(34,176,114,94),(35,189,115,107),(36,202,116,64),(37,215,117,77),(38,172,118,90),(39,185,119,103),(40,198,120,60),(41,211,121,73),(42,224,122,86),(43,181,123,99),(44,194,124,112),(45,207,125,69),(46,220,126,82),(47,177,127,95),(48,190,128,108),(49,203,129,65),(50,216,130,78),(51,173,131,91),(52,186,132,104),(53,199,133,61),(54,212,134,74),(55,169,135,87),(56,182,136,100)], [(2,42),(3,27),(4,12),(5,53),(6,38),(7,23),(9,49),(10,34),(11,19),(13,45),(14,30),(16,56),(17,41),(18,26),(20,52),(21,37),(24,48),(25,33),(28,44),(31,55),(32,40),(35,51),(39,47),(46,54),(57,195),(58,180),(59,221),(60,206),(61,191),(62,176),(63,217),(64,202),(65,187),(66,172),(67,213),(68,198),(69,183),(70,224),(71,209),(72,194),(73,179),(74,220),(75,205),(76,190),(77,175),(78,216),(79,201),(80,186),(81,171),(82,212),(83,197),(84,182),(85,223),(86,208),(87,193),(88,178),(89,219),(90,204),(91,189),(92,174),(93,215),(94,200),(95,185),(96,170),(97,211),(98,196),(99,181),(100,222),(101,207),(102,192),(103,177),(104,218),(105,203),(106,188),(107,173),(108,214),(109,199),(110,184),(111,169),(112,210),(113,161),(114,146),(115,131),(117,157),(118,142),(119,127),(120,168),(121,153),(122,138),(124,164),(125,149),(126,134),(128,160),(129,145),(132,156),(133,141),(135,167),(136,152),(139,163),(140,148),(143,159),(147,155),(150,166),(154,162)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28L | 28M | ··· | 28R | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 4 | 14 | 14 | 1 | 1 | 1 | 1 | 4 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | D7 | C4○D4 | M4(2) | D14 | D14 | C8○D4 | C7⋊D4 | C4×D7 | C4×D7 | C4○D28 | D7×M4(2) | D28.C4 |
| kernel | C56⋊D4 | Dic7⋊C8 | C56⋊C4 | D14⋊C8 | C28.55D4 | D7×C2×C8 | C4×C7⋊D4 | C14×M4(2) | Dic7⋊C4 | D14⋊C4 | C23.D7 | C2×C7⋊D4 | C56 | C2×M4(2) | C28 | D14 | C2×C8 | C22×C4 | C14 | C8 | C2×C4 | C23 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 2 | 4 | 6 | 3 | 4 | 12 | 6 | 6 | 12 | 6 | 6 |
Matrix representation of C56⋊D4 ►in GL4(𝔽113) generated by
| 34 | 1 | 0 | 0 |
| 53 | 88 | 0 | 0 |
| 0 | 0 | 11 | 91 |
| 0 | 0 | 87 | 102 |
| 12 | 85 | 0 | 0 |
| 98 | 101 | 0 | 0 |
| 0 | 0 | 28 | 51 |
| 0 | 0 | 91 | 85 |
| 0 | 104 | 0 | 0 |
| 25 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(113))| [34,53,0,0,1,88,0,0,0,0,11,87,0,0,91,102],[12,98,0,0,85,101,0,0,0,0,28,91,0,0,51,85],[0,25,0,0,104,0,0,0,0,0,1,0,0,0,0,1] >;
C56⋊D4 in GAP, Magma, Sage, TeX
C_{56}\rtimes D_4 % in TeX
G:=Group("C56:D4"); // GroupNames label
G:=SmallGroup(448,661);
// by ID
G=gap.SmallGroup(448,661);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,387,58,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=c^2=1,b*a*b^-1=a^13,c*a*c=a^41,c*b*c=b^-1>;
// generators/relations