metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C56⋊C4⋊12C2, C22⋊C8.8D7, Dic7⋊C8⋊17C2, (C8×Dic7)⋊14C2, C14.7(C8○D4), (C2×C8).193D14, C4⋊Dic7.10C4, C23.11(C4×D7), C23.D7.3C4, C2.9(D28.C4), (C22×C4).74D14, C28.296(C4○D4), (C2×C56).169C22, (C2×C28).818C23, C28.55D4.2C2, C4.122(D4⋊2D7), C2.9(D28.2C4), (C22×C28).91C22, C7⋊3(C42.7C22), C14.22(C42⋊C2), (C4×Dic7).270C22, C23.21D14.2C2, C2.10(C23.11D14), (C2×C4).30(C4×D7), (C2×C28).38(C2×C4), C22.102(C2×C4×D7), (C7×C22⋊C8).11C2, (C2×C7⋊C8).299C22, (C22×C14).36(C2×C4), (C2×C14).73(C22×C4), (C2×Dic7).16(C2×C4), (C2×C4).760(C22×D7), SmallGroup(448,254)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56⋊C4⋊C2
G = < a,b,c | a56=b4=c2=1, bab-1=a13, cac=ab2, cbc=a28b >
Subgroups: 316 in 96 conjugacy classes, 47 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, Dic7, C28, C28, C2×C14, C2×C14, C4×C8, C8⋊C4, C22⋊C8, C22⋊C8, C4⋊C8, C42⋊C2, C7⋊C8, C56, C2×Dic7, C2×C28, C2×C28, C22×C14, C42.7C22, C2×C7⋊C8, C4×Dic7, C4⋊Dic7, C23.D7, C2×C56, C22×C28, C8×Dic7, Dic7⋊C8, C56⋊C4, C28.55D4, C7×C22⋊C8, C23.21D14, C56⋊C4⋊C2
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C22×C4, C4○D4, D14, C42⋊C2, C8○D4, C4×D7, C22×D7, C42.7C22, C2×C4×D7, D4⋊2D7, C23.11D14, D28.2C4, D28.C4, C56⋊C4⋊C2
►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)
(1 115 65 190)(2 128 66 203)(3 141 67 216)(4 154 68 173)(5 167 69 186)(6 124 70 199)(7 137 71 212)(8 150 72 169)(9 163 73 182)(10 120 74 195)(11 133 75 208)(12 146 76 221)(13 159 77 178)(14 116 78 191)(15 129 79 204)(16 142 80 217)(17 155 81 174)(18 168 82 187)(19 125 83 200)(20 138 84 213)(21 151 85 170)(22 164 86 183)(23 121 87 196)(24 134 88 209)(25 147 89 222)(26 160 90 179)(27 117 91 192)(28 130 92 205)(29 143 93 218)(30 156 94 175)(31 113 95 188)(32 126 96 201)(33 139 97 214)(34 152 98 171)(35 165 99 184)(36 122 100 197)(37 135 101 210)(38 148 102 223)(39 161 103 180)(40 118 104 193)(41 131 105 206)(42 144 106 219)(43 157 107 176)(44 114 108 189)(45 127 109 202)(46 140 110 215)(47 153 111 172)(48 166 112 185)(49 123 57 198)(50 136 58 211)(51 149 59 224)(52 162 60 181)(53 119 61 194)(54 132 62 207)(55 145 63 220)(56 158 64 177)
(2 66)(4 68)(6 70)(8 72)(10 74)(12 76)(14 78)(16 80)(18 82)(20 84)(22 86)(24 88)(26 90)(28 92)(30 94)(32 96)(34 98)(36 100)(38 102)(40 104)(42 106)(44 108)(46 110)(48 112)(50 58)(52 60)(54 62)(56 64)(113 141)(114 217)(115 143)(116 219)(117 145)(118 221)(119 147)(120 223)(121 149)(122 169)(123 151)(124 171)(125 153)(126 173)(127 155)(128 175)(129 157)(130 177)(131 159)(132 179)(133 161)(134 181)(135 163)(136 183)(137 165)(138 185)(139 167)(140 187)(142 189)(144 191)(146 193)(148 195)(150 197)(152 199)(154 201)(156 203)(158 205)(160 207)(162 209)(164 211)(166 213)(168 215)(170 198)(172 200)(174 202)(176 204)(178 206)(180 208)(182 210)(184 212)(186 214)(188 216)(190 218)(192 220)(194 222)(196 224)
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), (1,115,65,190)(2,128,66,203)(3,141,67,216)(4,154,68,173)(5,167,69,186)(6,124,70,199)(7,137,71,212)(8,150,72,169)(9,163,73,182)(10,120,74,195)(11,133,75,208)(12,146,76,221)(13,159,77,178)(14,116,78,191)(15,129,79,204)(16,142,80,217)(17,155,81,174)(18,168,82,187)(19,125,83,200)(20,138,84,213)(21,151,85,170)(22,164,86,183)(23,121,87,196)(24,134,88,209)(25,147,89,222)(26,160,90,179)(27,117,91,192)(28,130,92,205)(29,143,93,218)(30,156,94,175)(31,113,95,188)(32,126,96,201)(33,139,97,214)(34,152,98,171)(35,165,99,184)(36,122,100,197)(37,135,101,210)(38,148,102,223)(39,161,103,180)(40,118,104,193)(41,131,105,206)(42,144,106,219)(43,157,107,176)(44,114,108,189)(45,127,109,202)(46,140,110,215)(47,153,111,172)(48,166,112,185)(49,123,57,198)(50,136,58,211)(51,149,59,224)(52,162,60,181)(53,119,61,194)(54,132,62,207)(55,145,63,220)(56,158,64,177), (2,66)(4,68)(6,70)(8,72)(10,74)(12,76)(14,78)(16,80)(18,82)(20,84)(22,86)(24,88)(26,90)(28,92)(30,94)(32,96)(34,98)(36,100)(38,102)(40,104)(42,106)(44,108)(46,110)(48,112)(50,58)(52,60)(54,62)(56,64)(113,141)(114,217)(115,143)(116,219)(117,145)(118,221)(119,147)(120,223)(121,149)(122,169)(123,151)(124,171)(125,153)(126,173)(127,155)(128,175)(129,157)(130,177)(131,159)(132,179)(133,161)(134,181)(135,163)(136,183)(137,165)(138,185)(139,167)(140,187)(142,189)(144,191)(146,193)(148,195)(150,197)(152,199)(154,201)(156,203)(158,205)(160,207)(162,209)(164,211)(166,213)(168,215)(170,198)(172,200)(174,202)(176,204)(178,206)(180,208)(182,210)(184,212)(186,214)(188,216)(190,218)(192,220)(194,222)(196,224)>;
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), (1,115,65,190)(2,128,66,203)(3,141,67,216)(4,154,68,173)(5,167,69,186)(6,124,70,199)(7,137,71,212)(8,150,72,169)(9,163,73,182)(10,120,74,195)(11,133,75,208)(12,146,76,221)(13,159,77,178)(14,116,78,191)(15,129,79,204)(16,142,80,217)(17,155,81,174)(18,168,82,187)(19,125,83,200)(20,138,84,213)(21,151,85,170)(22,164,86,183)(23,121,87,196)(24,134,88,209)(25,147,89,222)(26,160,90,179)(27,117,91,192)(28,130,92,205)(29,143,93,218)(30,156,94,175)(31,113,95,188)(32,126,96,201)(33,139,97,214)(34,152,98,171)(35,165,99,184)(36,122,100,197)(37,135,101,210)(38,148,102,223)(39,161,103,180)(40,118,104,193)(41,131,105,206)(42,144,106,219)(43,157,107,176)(44,114,108,189)(45,127,109,202)(46,140,110,215)(47,153,111,172)(48,166,112,185)(49,123,57,198)(50,136,58,211)(51,149,59,224)(52,162,60,181)(53,119,61,194)(54,132,62,207)(55,145,63,220)(56,158,64,177), (2,66)(4,68)(6,70)(8,72)(10,74)(12,76)(14,78)(16,80)(18,82)(20,84)(22,86)(24,88)(26,90)(28,92)(30,94)(32,96)(34,98)(36,100)(38,102)(40,104)(42,106)(44,108)(46,110)(48,112)(50,58)(52,60)(54,62)(56,64)(113,141)(114,217)(115,143)(116,219)(117,145)(118,221)(119,147)(120,223)(121,149)(122,169)(123,151)(124,171)(125,153)(126,173)(127,155)(128,175)(129,157)(130,177)(131,159)(132,179)(133,161)(134,181)(135,163)(136,183)(137,165)(138,185)(139,167)(140,187)(142,189)(144,191)(146,193)(148,195)(150,197)(152,199)(154,201)(156,203)(158,205)(160,207)(162,209)(164,211)(166,213)(168,215)(170,198)(172,200)(174,202)(176,204)(178,206)(180,208)(182,210)(184,212)(186,214)(188,216)(190,218)(192,220)(194,222)(196,224) );
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)], [(1,115,65,190),(2,128,66,203),(3,141,67,216),(4,154,68,173),(5,167,69,186),(6,124,70,199),(7,137,71,212),(8,150,72,169),(9,163,73,182),(10,120,74,195),(11,133,75,208),(12,146,76,221),(13,159,77,178),(14,116,78,191),(15,129,79,204),(16,142,80,217),(17,155,81,174),(18,168,82,187),(19,125,83,200),(20,138,84,213),(21,151,85,170),(22,164,86,183),(23,121,87,196),(24,134,88,209),(25,147,89,222),(26,160,90,179),(27,117,91,192),(28,130,92,205),(29,143,93,218),(30,156,94,175),(31,113,95,188),(32,126,96,201),(33,139,97,214),(34,152,98,171),(35,165,99,184),(36,122,100,197),(37,135,101,210),(38,148,102,223),(39,161,103,180),(40,118,104,193),(41,131,105,206),(42,144,106,219),(43,157,107,176),(44,114,108,189),(45,127,109,202),(46,140,110,215),(47,153,111,172),(48,166,112,185),(49,123,57,198),(50,136,58,211),(51,149,59,224),(52,162,60,181),(53,119,61,194),(54,132,62,207),(55,145,63,220),(56,158,64,177)], [(2,66),(4,68),(6,70),(8,72),(10,74),(12,76),(14,78),(16,80),(18,82),(20,84),(22,86),(24,88),(26,90),(28,92),(30,94),(32,96),(34,98),(36,100),(38,102),(40,104),(42,106),(44,108),(46,110),(48,112),(50,58),(52,60),(54,62),(56,64),(113,141),(114,217),(115,143),(116,219),(117,145),(118,221),(119,147),(120,223),(121,149),(122,169),(123,151),(124,171),(125,153),(126,173),(127,155),(128,175),(129,157),(130,177),(131,159),(132,179),(133,161),(134,181),(135,163),(136,183),(137,165),(138,185),(139,167),(140,187),(142,189),(144,191),(146,193),(148,195),(150,197),(152,199),(154,201),(156,203),(158,205),(160,207),(162,209),(164,211),(166,213),(168,215),(170,198),(172,200),(174,202),(176,204),(178,206),(180,208),(182,210),(184,212),(186,214),(188,216),(190,218),(192,220),(194,222),(196,224)]])
88 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 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 | 4 | 4 | 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 | 1 | 1 | 1 | 1 | 4 | 14 | 14 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D7 | C4○D4 | D14 | D14 | C8○D4 | C4×D7 | C4×D7 | D28.2C4 | D4⋊2D7 | D28.C4 |
| kernel | C56⋊C4⋊C2 | C8×Dic7 | Dic7⋊C8 | C56⋊C4 | C28.55D4 | C7×C22⋊C8 | C23.21D14 | C4⋊Dic7 | C23.D7 | C22⋊C8 | C28 | C2×C8 | C22×C4 | C14 | C2×C4 | C23 | C2 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 4 | 3 | 4 | 6 | 3 | 8 | 6 | 6 | 24 | 6 | 6 |
Matrix representation of C56⋊C4⋊C2 ►in GL6(𝔽113)
| 69 | 0 | 0 | 0 | 0 | 0 |
| 0 | 44 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 3 |
| 0 | 0 | 0 | 0 | 0 | 14 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 0 | 0 | 0 |
| 0 | 0 | 0 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 45 |
| 0 | 0 | 0 | 0 | 14 | 106 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(113))| [69,0,0,0,0,0,0,44,0,0,0,0,0,0,0,112,0,0,0,0,112,0,0,0,0,0,0,0,8,0,0,0,0,0,3,14],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,15,0,0,0,0,0,0,15,0,0,0,0,0,0,7,14,0,0,0,0,45,106],[1,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C56⋊C4⋊C2 in GAP, Magma, Sage, TeX
C_{56}\rtimes C_4\rtimes C_2 % in TeX
G:=Group("C56:C4:C2"); // GroupNames label
G:=SmallGroup(448,254);
// by ID
G=gap.SmallGroup(448,254);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,477,219,58,136,18822]);
// Polycyclic
G:=Group<a,b,c|a^56=b^4=c^2=1,b*a*b^-1=a^13,c*a*c=a*b^2,c*b*c=a^28*b>;
// generators/relations