metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic44, C8.D11, C11⋊1Q16, C88.1C2, C22.3D4, C2.5D44, C4.10D22, C44.10C22, Dic22.1C2, SmallGroup(176,7)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic44
G = < a,b | a88=1, b2=a44, bab-1=a-1 >
Smallest permutation representation of Dic44
►Regular action 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)
(1 152 45 108)(2 151 46 107)(3 150 47 106)(4 149 48 105)(5 148 49 104)(6 147 50 103)(7 146 51 102)(8 145 52 101)(9 144 53 100)(10 143 54 99)(11 142 55 98)(12 141 56 97)(13 140 57 96)(14 139 58 95)(15 138 59 94)(16 137 60 93)(17 136 61 92)(18 135 62 91)(19 134 63 90)(20 133 64 89)(21 132 65 176)(22 131 66 175)(23 130 67 174)(24 129 68 173)(25 128 69 172)(26 127 70 171)(27 126 71 170)(28 125 72 169)(29 124 73 168)(30 123 74 167)(31 122 75 166)(32 121 76 165)(33 120 77 164)(34 119 78 163)(35 118 79 162)(36 117 80 161)(37 116 81 160)(38 115 82 159)(39 114 83 158)(40 113 84 157)(41 112 85 156)(42 111 86 155)(43 110 87 154)(44 109 88 153)
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), (1,152,45,108)(2,151,46,107)(3,150,47,106)(4,149,48,105)(5,148,49,104)(6,147,50,103)(7,146,51,102)(8,145,52,101)(9,144,53,100)(10,143,54,99)(11,142,55,98)(12,141,56,97)(13,140,57,96)(14,139,58,95)(15,138,59,94)(16,137,60,93)(17,136,61,92)(18,135,62,91)(19,134,63,90)(20,133,64,89)(21,132,65,176)(22,131,66,175)(23,130,67,174)(24,129,68,173)(25,128,69,172)(26,127,70,171)(27,126,71,170)(28,125,72,169)(29,124,73,168)(30,123,74,167)(31,122,75,166)(32,121,76,165)(33,120,77,164)(34,119,78,163)(35,118,79,162)(36,117,80,161)(37,116,81,160)(38,115,82,159)(39,114,83,158)(40,113,84,157)(41,112,85,156)(42,111,86,155)(43,110,87,154)(44,109,88,153)>;
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), (1,152,45,108)(2,151,46,107)(3,150,47,106)(4,149,48,105)(5,148,49,104)(6,147,50,103)(7,146,51,102)(8,145,52,101)(9,144,53,100)(10,143,54,99)(11,142,55,98)(12,141,56,97)(13,140,57,96)(14,139,58,95)(15,138,59,94)(16,137,60,93)(17,136,61,92)(18,135,62,91)(19,134,63,90)(20,133,64,89)(21,132,65,176)(22,131,66,175)(23,130,67,174)(24,129,68,173)(25,128,69,172)(26,127,70,171)(27,126,71,170)(28,125,72,169)(29,124,73,168)(30,123,74,167)(31,122,75,166)(32,121,76,165)(33,120,77,164)(34,119,78,163)(35,118,79,162)(36,117,80,161)(37,116,81,160)(38,115,82,159)(39,114,83,158)(40,113,84,157)(41,112,85,156)(42,111,86,155)(43,110,87,154)(44,109,88,153) );
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)], [(1,152,45,108),(2,151,46,107),(3,150,47,106),(4,149,48,105),(5,148,49,104),(6,147,50,103),(7,146,51,102),(8,145,52,101),(9,144,53,100),(10,143,54,99),(11,142,55,98),(12,141,56,97),(13,140,57,96),(14,139,58,95),(15,138,59,94),(16,137,60,93),(17,136,61,92),(18,135,62,91),(19,134,63,90),(20,133,64,89),(21,132,65,176),(22,131,66,175),(23,130,67,174),(24,129,68,173),(25,128,69,172),(26,127,70,171),(27,126,71,170),(28,125,72,169),(29,124,73,168),(30,123,74,167),(31,122,75,166),(32,121,76,165),(33,120,77,164),(34,119,78,163),(35,118,79,162),(36,117,80,161),(37,116,81,160),(38,115,82,159),(39,114,83,158),(40,113,84,157),(41,112,85,156),(42,111,86,155),(43,110,87,154),(44,109,88,153)]])
Dic44 is a maximal subgroup of
C176⋊C2 Dic88 D8.D11 C11⋊Q32 D88⋊7C2 C8.D22 D8⋊3D11 D4.D22 Q16×D11
Dic44 is a maximal quotient of
C44.44D4 C44.5Q8
47 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 8A | 8B | 11A | ··· | 11E | 22A | ··· | 22E | 44A | ··· | 44J | 88A | ··· | 88T |
| order | 1 | 2 | 4 | 4 | 4 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 2 | 44 | 44 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
47 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | + | - |
| image | C1 | C2 | C2 | D4 | Q16 | D11 | D22 | D44 | Dic44 |
| kernel | Dic44 | C88 | Dic22 | C22 | C11 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 5 | 5 | 10 | 20 |
Matrix representation of Dic44 ►in GL2(𝔽89) generated by
| 5 | 82 |
| 7 | 26 |
| 54 | 15 |
| 31 | 35 |
G:=sub<GL(2,GF(89))| [5,7,82,26],[54,31,15,35] >;
Dic44 in GAP, Magma, Sage, TeX
{\rm Dic}_{44} % in TeX
G:=Group("Dic44"); // GroupNames label
G:=SmallGroup(176,7);
// by ID
G=gap.SmallGroup(176,7);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-11,40,61,66,182,42,4004]);
// Polycyclic
G:=Group<a,b|a^88=1,b^2=a^44,b*a*b^-1=a^-1>;
// generators/relations
Export