direct product, abelian, monomial, 2-elementary
Aliases: C4×C44, SmallGroup(176,19)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C4×C44 |
| C1 — C4×C44 |
| C1 — C4×C44 |
Generators and relations for C4×C44
G = < a,b | a4=b44=1, ab=ba >
Smallest permutation representation of C4×C44
►Regular action on 176 points
Generators in S176
(1 66 100 157)(2 67 101 158)(3 68 102 159)(4 69 103 160)(5 70 104 161)(6 71 105 162)(7 72 106 163)(8 73 107 164)(9 74 108 165)(10 75 109 166)(11 76 110 167)(12 77 111 168)(13 78 112 169)(14 79 113 170)(15 80 114 171)(16 81 115 172)(17 82 116 173)(18 83 117 174)(19 84 118 175)(20 85 119 176)(21 86 120 133)(22 87 121 134)(23 88 122 135)(24 45 123 136)(25 46 124 137)(26 47 125 138)(27 48 126 139)(28 49 127 140)(29 50 128 141)(30 51 129 142)(31 52 130 143)(32 53 131 144)(33 54 132 145)(34 55 89 146)(35 56 90 147)(36 57 91 148)(37 58 92 149)(38 59 93 150)(39 60 94 151)(40 61 95 152)(41 62 96 153)(42 63 97 154)(43 64 98 155)(44 65 99 156)
(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)
G:=sub<Sym(176)| (1,66,100,157)(2,67,101,158)(3,68,102,159)(4,69,103,160)(5,70,104,161)(6,71,105,162)(7,72,106,163)(8,73,107,164)(9,74,108,165)(10,75,109,166)(11,76,110,167)(12,77,111,168)(13,78,112,169)(14,79,113,170)(15,80,114,171)(16,81,115,172)(17,82,116,173)(18,83,117,174)(19,84,118,175)(20,85,119,176)(21,86,120,133)(22,87,121,134)(23,88,122,135)(24,45,123,136)(25,46,124,137)(26,47,125,138)(27,48,126,139)(28,49,127,140)(29,50,128,141)(30,51,129,142)(31,52,130,143)(32,53,131,144)(33,54,132,145)(34,55,89,146)(35,56,90,147)(36,57,91,148)(37,58,92,149)(38,59,93,150)(39,60,94,151)(40,61,95,152)(41,62,96,153)(42,63,97,154)(43,64,98,155)(44,65,99,156), (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)>;
G:=Group( (1,66,100,157)(2,67,101,158)(3,68,102,159)(4,69,103,160)(5,70,104,161)(6,71,105,162)(7,72,106,163)(8,73,107,164)(9,74,108,165)(10,75,109,166)(11,76,110,167)(12,77,111,168)(13,78,112,169)(14,79,113,170)(15,80,114,171)(16,81,115,172)(17,82,116,173)(18,83,117,174)(19,84,118,175)(20,85,119,176)(21,86,120,133)(22,87,121,134)(23,88,122,135)(24,45,123,136)(25,46,124,137)(26,47,125,138)(27,48,126,139)(28,49,127,140)(29,50,128,141)(30,51,129,142)(31,52,130,143)(32,53,131,144)(33,54,132,145)(34,55,89,146)(35,56,90,147)(36,57,91,148)(37,58,92,149)(38,59,93,150)(39,60,94,151)(40,61,95,152)(41,62,96,153)(42,63,97,154)(43,64,98,155)(44,65,99,156), (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) );
G=PermutationGroup([[(1,66,100,157),(2,67,101,158),(3,68,102,159),(4,69,103,160),(5,70,104,161),(6,71,105,162),(7,72,106,163),(8,73,107,164),(9,74,108,165),(10,75,109,166),(11,76,110,167),(12,77,111,168),(13,78,112,169),(14,79,113,170),(15,80,114,171),(16,81,115,172),(17,82,116,173),(18,83,117,174),(19,84,118,175),(20,85,119,176),(21,86,120,133),(22,87,121,134),(23,88,122,135),(24,45,123,136),(25,46,124,137),(26,47,125,138),(27,48,126,139),(28,49,127,140),(29,50,128,141),(30,51,129,142),(31,52,130,143),(32,53,131,144),(33,54,132,145),(34,55,89,146),(35,56,90,147),(36,57,91,148),(37,58,92,149),(38,59,93,150),(39,60,94,151),(40,61,95,152),(41,62,96,153),(42,63,97,154),(43,64,98,155),(44,65,99,156)], [(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)]])
C4×C44 is a maximal subgroup of
C42.D11 C44⋊C8 D44⋊1C4 C44⋊2Q8 C44.6Q8 C42⋊D11 C4⋊D44 C4.D44 C42⋊2D11
176 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 11A | ··· | 11J | 22A | ··· | 22AD | 44A | ··· | 44DP |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
176 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||
| image | C1 | C2 | C4 | C11 | C22 | C44 |
| kernel | C4×C44 | C2×C44 | C44 | C42 | C2×C4 | C4 |
| # reps | 1 | 3 | 12 | 10 | 30 | 120 |
Matrix representation of C4×C44 ►in GL2(𝔽89) generated by
| 55 | 0 |
| 0 | 1 |
| 72 | 0 |
| 0 | 20 |
G:=sub<GL(2,GF(89))| [55,0,0,1],[72,0,0,20] >;
C4×C44 in GAP, Magma, Sage, TeX
C_4\times C_{44} % in TeX
G:=Group("C4xC44"); // GroupNames label
G:=SmallGroup(176,19);
// by ID
G=gap.SmallGroup(176,19);
# by ID
G:=PCGroup([5,-2,-2,-11,-2,-2,220,446]);
// Polycyclic
G:=Group<a,b|a^4=b^44=1,a*b=b*a>;
// generators/relations
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