metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C88.1C4, C4.18D44, C44.34D4, C8.1Dic11, C22.2Dic22, (C2×C88).7C2, (C2×C8).5D11, C22.7(C4⋊C4), (C2×C22).3Q8, C44.35(C2×C4), (C2×C4).70D22, C11⋊1(C8.C4), C4.8(C2×Dic11), C2.5(C44⋊C4), C44.C4.1C2, (C2×C44).97C22, SmallGroup(352,25)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C88.C4
G = < a,b | a88=1, b4=a44, bab-1=a43 >
Smallest permutation representation of C88.C4
►On 176 points
Generators in S176
(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)
(1 131 23 109 45 175 67 153)(2 174 24 152 46 130 68 108)(3 129 25 107 47 173 69 151)(4 172 26 150 48 128 70 106)(5 127 27 105 49 171 71 149)(6 170 28 148 50 126 72 104)(7 125 29 103 51 169 73 147)(8 168 30 146 52 124 74 102)(9 123 31 101 53 167 75 145)(10 166 32 144 54 122 76 100)(11 121 33 99 55 165 77 143)(12 164 34 142 56 120 78 98)(13 119 35 97 57 163 79 141)(14 162 36 140 58 118 80 96)(15 117 37 95 59 161 81 139)(16 160 38 138 60 116 82 94)(17 115 39 93 61 159 83 137)(18 158 40 136 62 114 84 92)(19 113 41 91 63 157 85 135)(20 156 42 134 64 112 86 90)(21 111 43 89 65 155 87 133)(22 154 44 132 66 110 88 176)
G:=sub<Sym(176)| (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), (1,131,23,109,45,175,67,153)(2,174,24,152,46,130,68,108)(3,129,25,107,47,173,69,151)(4,172,26,150,48,128,70,106)(5,127,27,105,49,171,71,149)(6,170,28,148,50,126,72,104)(7,125,29,103,51,169,73,147)(8,168,30,146,52,124,74,102)(9,123,31,101,53,167,75,145)(10,166,32,144,54,122,76,100)(11,121,33,99,55,165,77,143)(12,164,34,142,56,120,78,98)(13,119,35,97,57,163,79,141)(14,162,36,140,58,118,80,96)(15,117,37,95,59,161,81,139)(16,160,38,138,60,116,82,94)(17,115,39,93,61,159,83,137)(18,158,40,136,62,114,84,92)(19,113,41,91,63,157,85,135)(20,156,42,134,64,112,86,90)(21,111,43,89,65,155,87,133)(22,154,44,132,66,110,88,176)>;
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), (1,131,23,109,45,175,67,153)(2,174,24,152,46,130,68,108)(3,129,25,107,47,173,69,151)(4,172,26,150,48,128,70,106)(5,127,27,105,49,171,71,149)(6,170,28,148,50,126,72,104)(7,125,29,103,51,169,73,147)(8,168,30,146,52,124,74,102)(9,123,31,101,53,167,75,145)(10,166,32,144,54,122,76,100)(11,121,33,99,55,165,77,143)(12,164,34,142,56,120,78,98)(13,119,35,97,57,163,79,141)(14,162,36,140,58,118,80,96)(15,117,37,95,59,161,81,139)(16,160,38,138,60,116,82,94)(17,115,39,93,61,159,83,137)(18,158,40,136,62,114,84,92)(19,113,41,91,63,157,85,135)(20,156,42,134,64,112,86,90)(21,111,43,89,65,155,87,133)(22,154,44,132,66,110,88,176) );
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)], [(1,131,23,109,45,175,67,153),(2,174,24,152,46,130,68,108),(3,129,25,107,47,173,69,151),(4,172,26,150,48,128,70,106),(5,127,27,105,49,171,71,149),(6,170,28,148,50,126,72,104),(7,125,29,103,51,169,73,147),(8,168,30,146,52,124,74,102),(9,123,31,101,53,167,75,145),(10,166,32,144,54,122,76,100),(11,121,33,99,55,165,77,143),(12,164,34,142,56,120,78,98),(13,119,35,97,57,163,79,141),(14,162,36,140,58,118,80,96),(15,117,37,95,59,161,81,139),(16,160,38,138,60,116,82,94),(17,115,39,93,61,159,83,137),(18,158,40,136,62,114,84,92),(19,113,41,91,63,157,85,135),(20,156,42,134,64,112,86,90),(21,111,43,89,65,155,87,133),(22,154,44,132,66,110,88,176)]])
94 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 11A | ··· | 11E | 22A | ··· | 22O | 44A | ··· | 44T | 88A | ··· | 88AN |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 44 | 44 | 44 | 44 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
94 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C4 | D4 | Q8 | D11 | C8.C4 | Dic11 | D22 | D44 | Dic22 | C88.C4 |
| kernel | C88.C4 | C44.C4 | C2×C88 | C88 | C44 | C2×C22 | C2×C8 | C11 | C8 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 5 | 4 | 10 | 5 | 10 | 10 | 40 |
Matrix representation of C88.C4 ►in GL2(𝔽89) generated by
| 29 | 0 |
| 25 | 46 |
| 31 | 46 |
| 19 | 58 |
G:=sub<GL(2,GF(89))| [29,25,0,46],[31,19,46,58] >;
C88.C4 in GAP, Magma, Sage, TeX
C_{88}.C_4 % in TeX
G:=Group("C88.C4"); // GroupNames label
G:=SmallGroup(352,25);
// by ID
G=gap.SmallGroup(352,25);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,24,121,55,86,579,69,11525]);
// Polycyclic
G:=Group<a,b|a^88=1,b^4=a^44,b*a*b^-1=a^43>;
// generators/relations
Export