metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C56⋊1C4, C8⋊1Dic7, C14.4D8, C2.1D56, C28.5Q8, C14.2Q16, C4.5Dic14, C2.2Dic28, C22.9D28, (C2×C8).3D7, C7⋊2(C2.D8), (C2×C56).5C2, C14.6(C4⋊C4), C28.34(C2×C4), (C2×C14).14D4, (C2×C4).69D14, C4⋊Dic7.3C2, C4.7(C2×Dic7), C2.4(C4⋊Dic7), (C2×C28).82C22, SmallGroup(224,24)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C56⋊1C4
G = < a,b | a56=b4=1, bab-1=a-1 >
Smallest permutation representation of C56⋊1C4
►Regular action 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 61 186 128)(2 60 187 127)(3 59 188 126)(4 58 189 125)(5 57 190 124)(6 112 191 123)(7 111 192 122)(8 110 193 121)(9 109 194 120)(10 108 195 119)(11 107 196 118)(12 106 197 117)(13 105 198 116)(14 104 199 115)(15 103 200 114)(16 102 201 113)(17 101 202 168)(18 100 203 167)(19 99 204 166)(20 98 205 165)(21 97 206 164)(22 96 207 163)(23 95 208 162)(24 94 209 161)(25 93 210 160)(26 92 211 159)(27 91 212 158)(28 90 213 157)(29 89 214 156)(30 88 215 155)(31 87 216 154)(32 86 217 153)(33 85 218 152)(34 84 219 151)(35 83 220 150)(36 82 221 149)(37 81 222 148)(38 80 223 147)(39 79 224 146)(40 78 169 145)(41 77 170 144)(42 76 171 143)(43 75 172 142)(44 74 173 141)(45 73 174 140)(46 72 175 139)(47 71 176 138)(48 70 177 137)(49 69 178 136)(50 68 179 135)(51 67 180 134)(52 66 181 133)(53 65 182 132)(54 64 183 131)(55 63 184 130)(56 62 185 129)
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), (1,61,186,128)(2,60,187,127)(3,59,188,126)(4,58,189,125)(5,57,190,124)(6,112,191,123)(7,111,192,122)(8,110,193,121)(9,109,194,120)(10,108,195,119)(11,107,196,118)(12,106,197,117)(13,105,198,116)(14,104,199,115)(15,103,200,114)(16,102,201,113)(17,101,202,168)(18,100,203,167)(19,99,204,166)(20,98,205,165)(21,97,206,164)(22,96,207,163)(23,95,208,162)(24,94,209,161)(25,93,210,160)(26,92,211,159)(27,91,212,158)(28,90,213,157)(29,89,214,156)(30,88,215,155)(31,87,216,154)(32,86,217,153)(33,85,218,152)(34,84,219,151)(35,83,220,150)(36,82,221,149)(37,81,222,148)(38,80,223,147)(39,79,224,146)(40,78,169,145)(41,77,170,144)(42,76,171,143)(43,75,172,142)(44,74,173,141)(45,73,174,140)(46,72,175,139)(47,71,176,138)(48,70,177,137)(49,69,178,136)(50,68,179,135)(51,67,180,134)(52,66,181,133)(53,65,182,132)(54,64,183,131)(55,63,184,130)(56,62,185,129)>;
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), (1,61,186,128)(2,60,187,127)(3,59,188,126)(4,58,189,125)(5,57,190,124)(6,112,191,123)(7,111,192,122)(8,110,193,121)(9,109,194,120)(10,108,195,119)(11,107,196,118)(12,106,197,117)(13,105,198,116)(14,104,199,115)(15,103,200,114)(16,102,201,113)(17,101,202,168)(18,100,203,167)(19,99,204,166)(20,98,205,165)(21,97,206,164)(22,96,207,163)(23,95,208,162)(24,94,209,161)(25,93,210,160)(26,92,211,159)(27,91,212,158)(28,90,213,157)(29,89,214,156)(30,88,215,155)(31,87,216,154)(32,86,217,153)(33,85,218,152)(34,84,219,151)(35,83,220,150)(36,82,221,149)(37,81,222,148)(38,80,223,147)(39,79,224,146)(40,78,169,145)(41,77,170,144)(42,76,171,143)(43,75,172,142)(44,74,173,141)(45,73,174,140)(46,72,175,139)(47,71,176,138)(48,70,177,137)(49,69,178,136)(50,68,179,135)(51,67,180,134)(52,66,181,133)(53,65,182,132)(54,64,183,131)(55,63,184,130)(56,62,185,129) );
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)], [(1,61,186,128),(2,60,187,127),(3,59,188,126),(4,58,189,125),(5,57,190,124),(6,112,191,123),(7,111,192,122),(8,110,193,121),(9,109,194,120),(10,108,195,119),(11,107,196,118),(12,106,197,117),(13,105,198,116),(14,104,199,115),(15,103,200,114),(16,102,201,113),(17,101,202,168),(18,100,203,167),(19,99,204,166),(20,98,205,165),(21,97,206,164),(22,96,207,163),(23,95,208,162),(24,94,209,161),(25,93,210,160),(26,92,211,159),(27,91,212,158),(28,90,213,157),(29,89,214,156),(30,88,215,155),(31,87,216,154),(32,86,217,153),(33,85,218,152),(34,84,219,151),(35,83,220,150),(36,82,221,149),(37,81,222,148),(38,80,223,147),(39,79,224,146),(40,78,169,145),(41,77,170,144),(42,76,171,143),(43,75,172,142),(44,74,173,141),(45,73,174,140),(46,72,175,139),(47,71,176,138),(48,70,177,137),(49,69,178,136),(50,68,179,135),(51,67,180,134),(52,66,181,133),(53,65,182,132),(54,64,183,131),(55,63,184,130),(56,62,185,129)]])
C56⋊1C4 is a maximal subgroup of
C8.4Dic14 C8.5Dic14 C56.78D4 C112⋊5C4 C112⋊6C4 C2.D112 C14.SD32 C14.Q32 C56⋊8Q8 C56.13Q8 C4×D56 C4×Dic28 C8⋊Dic14 C42.16D14 C23.35D28 C23.10D28 C22.D56 C23.13D28 Dic7.D8 D4.2Dic14 D14.D8 C56⋊1C4⋊C2 Dic7.Q16 Q8.Dic14 D14.Q16 D14⋊C8.C2 Dic14.3Q8 D28⋊4Q8 D28.3Q8 C28.7Q16 C56⋊3Q8 C8⋊(C4×D7) C56⋊2Q8 C56.4Q8 D7×C2.D8 C8.27(C4×D7) C23.22D28 C56⋊29D4 C56.82D4 C23.47D28 C56⋊2D4 D8×Dic7 C56⋊6D4 SD16⋊Dic7 C56⋊8D4 Q16×Dic7 D14⋊3Q16
C56⋊1C4 is a maximal quotient of
C56⋊1C8 C112⋊5C4 C112⋊6C4 C112.C4 C28.9C42
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 28A | ··· | 28L | 56A | ··· | 56X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | + | - | - | + | - | + | + | - | |
| image | C1 | C2 | C2 | C4 | Q8 | D4 | D7 | D8 | Q16 | Dic7 | D14 | Dic14 | D28 | D56 | Dic28 |
| kernel | C56⋊1C4 | C4⋊Dic7 | C2×C56 | C56 | C28 | C2×C14 | C2×C8 | C14 | C14 | C8 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 3 | 2 | 2 | 6 | 3 | 6 | 6 | 12 | 12 |
Matrix representation of C56⋊1C4 ►in GL4(𝔽113) generated by
| 38 | 55 | 0 | 0 |
| 58 | 81 | 0 | 0 |
| 0 | 0 | 48 | 41 |
| 0 | 0 | 72 | 78 |
| 31 | 57 | 0 | 0 |
| 110 | 82 | 0 | 0 |
| 0 | 0 | 21 | 65 |
| 0 | 0 | 28 | 92 |
G:=sub<GL(4,GF(113))| [38,58,0,0,55,81,0,0,0,0,48,72,0,0,41,78],[31,110,0,0,57,82,0,0,0,0,21,28,0,0,65,92] >;
C56⋊1C4 in GAP, Magma, Sage, TeX
C_{56}\rtimes_1C_4 % in TeX
G:=Group("C56:1C4"); // GroupNames label
G:=SmallGroup(224,24);
// by ID
G=gap.SmallGroup(224,24);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,24,121,151,579,69,6917]);
// Polycyclic
G:=Group<a,b|a^56=b^4=1,b*a*b^-1=a^-1>;
// generators/relations
Export