metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C112.1C4, C28.36D8, C4.18D56, C56.14Q8, C16.1Dic7, C8.13Dic14, C22.1Dic28, (C2×C16).5D7, C56.71(C2×C4), (C2×C112).7C2, (C2×C4).73D28, C28.25(C4⋊C4), (C2×C14).7Q16, C7⋊2(C8.4Q8), (C2×C8).311D14, (C2×C28).393D4, C8.16(C2×Dic7), C2.5(C56⋊1C4), C14.9(C2.D8), C56.C4.1C2, C4.10(C4⋊Dic7), (C2×C56).383C22, SmallGroup(448,63)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C112.C4
G = < a,b | a112=1, b4=a56, bab-1=a55 >
Smallest permutation representation of C112.C4
►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)
(1 167 29 139 57 223 85 195)(2 222 30 194 58 166 86 138)(3 165 31 137 59 221 87 193)(4 220 32 192 60 164 88 136)(5 163 33 135 61 219 89 191)(6 218 34 190 62 162 90 134)(7 161 35 133 63 217 91 189)(8 216 36 188 64 160 92 132)(9 159 37 131 65 215 93 187)(10 214 38 186 66 158 94 130)(11 157 39 129 67 213 95 185)(12 212 40 184 68 156 96 128)(13 155 41 127 69 211 97 183)(14 210 42 182 70 154 98 126)(15 153 43 125 71 209 99 181)(16 208 44 180 72 152 100 124)(17 151 45 123 73 207 101 179)(18 206 46 178 74 150 102 122)(19 149 47 121 75 205 103 177)(20 204 48 176 76 148 104 120)(21 147 49 119 77 203 105 175)(22 202 50 174 78 146 106 118)(23 145 51 117 79 201 107 173)(24 200 52 172 80 144 108 116)(25 143 53 115 81 199 109 171)(26 198 54 170 82 142 110 114)(27 141 55 113 83 197 111 169)(28 196 56 168 84 140 112 224)
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), (1,167,29,139,57,223,85,195)(2,222,30,194,58,166,86,138)(3,165,31,137,59,221,87,193)(4,220,32,192,60,164,88,136)(5,163,33,135,61,219,89,191)(6,218,34,190,62,162,90,134)(7,161,35,133,63,217,91,189)(8,216,36,188,64,160,92,132)(9,159,37,131,65,215,93,187)(10,214,38,186,66,158,94,130)(11,157,39,129,67,213,95,185)(12,212,40,184,68,156,96,128)(13,155,41,127,69,211,97,183)(14,210,42,182,70,154,98,126)(15,153,43,125,71,209,99,181)(16,208,44,180,72,152,100,124)(17,151,45,123,73,207,101,179)(18,206,46,178,74,150,102,122)(19,149,47,121,75,205,103,177)(20,204,48,176,76,148,104,120)(21,147,49,119,77,203,105,175)(22,202,50,174,78,146,106,118)(23,145,51,117,79,201,107,173)(24,200,52,172,80,144,108,116)(25,143,53,115,81,199,109,171)(26,198,54,170,82,142,110,114)(27,141,55,113,83,197,111,169)(28,196,56,168,84,140,112,224)>;
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), (1,167,29,139,57,223,85,195)(2,222,30,194,58,166,86,138)(3,165,31,137,59,221,87,193)(4,220,32,192,60,164,88,136)(5,163,33,135,61,219,89,191)(6,218,34,190,62,162,90,134)(7,161,35,133,63,217,91,189)(8,216,36,188,64,160,92,132)(9,159,37,131,65,215,93,187)(10,214,38,186,66,158,94,130)(11,157,39,129,67,213,95,185)(12,212,40,184,68,156,96,128)(13,155,41,127,69,211,97,183)(14,210,42,182,70,154,98,126)(15,153,43,125,71,209,99,181)(16,208,44,180,72,152,100,124)(17,151,45,123,73,207,101,179)(18,206,46,178,74,150,102,122)(19,149,47,121,75,205,103,177)(20,204,48,176,76,148,104,120)(21,147,49,119,77,203,105,175)(22,202,50,174,78,146,106,118)(23,145,51,117,79,201,107,173)(24,200,52,172,80,144,108,116)(25,143,53,115,81,199,109,171)(26,198,54,170,82,142,110,114)(27,141,55,113,83,197,111,169)(28,196,56,168,84,140,112,224) );
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)], [(1,167,29,139,57,223,85,195),(2,222,30,194,58,166,86,138),(3,165,31,137,59,221,87,193),(4,220,32,192,60,164,88,136),(5,163,33,135,61,219,89,191),(6,218,34,190,62,162,90,134),(7,161,35,133,63,217,91,189),(8,216,36,188,64,160,92,132),(9,159,37,131,65,215,93,187),(10,214,38,186,66,158,94,130),(11,157,39,129,67,213,95,185),(12,212,40,184,68,156,96,128),(13,155,41,127,69,211,97,183),(14,210,42,182,70,154,98,126),(15,153,43,125,71,209,99,181),(16,208,44,180,72,152,100,124),(17,151,45,123,73,207,101,179),(18,206,46,178,74,150,102,122),(19,149,47,121,75,205,103,177),(20,204,48,176,76,148,104,120),(21,147,49,119,77,203,105,175),(22,202,50,174,78,146,106,118),(23,145,51,117,79,201,107,173),(24,200,52,172,80,144,108,116),(25,143,53,115,81,199,109,171),(26,198,54,170,82,142,110,114),(27,141,55,113,83,197,111,169),(28,196,56,168,84,140,112,224)]])
118 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | ··· | 14I | 16A | ··· | 16H | 28A | ··· | 28L | 56A | ··· | 56X | 112A | ··· | 112AV |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 56 | 56 | 56 | 56 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
118 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 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 | C8.4Q8 | D56 | Dic28 | C112.C4 |
| kernel | C112.C4 | C56.C4 | C2×C112 | C112 | C56 | C2×C28 | C2×C16 | C28 | C2×C14 | C16 | C2×C8 | C8 | C2×C4 | C7 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 3 | 2 | 2 | 6 | 3 | 6 | 6 | 8 | 12 | 12 | 48 |
Matrix representation of C112.C4 ►in GL2(𝔽113) generated by
| 68 | 0 |
| 46 | 108 |
| 9 | 111 |
| 48 | 104 |
G:=sub<GL(2,GF(113))| [68,46,0,108],[9,48,111,104] >;
C112.C4 in GAP, Magma, Sage, TeX
C_{112}.C_4 % in TeX
G:=Group("C112.C4"); // GroupNames label
G:=SmallGroup(448,63);
// by ID
G=gap.SmallGroup(448,63);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,176,184,675,192,1684,102,18822]);
// Polycyclic
G:=Group<a,b|a^112=1,b^4=a^56,b*a*b^-1=a^55>;
// generators/relations
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