metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D44⋊2C4, C2.2D88, C22.5D8, C44.45D4, C22.3SD16, C22.10D44, (C2×C88)⋊2C2, (C2×C8)⋊2D11, C44⋊C4⋊1C2, C4.8(C4×D11), C44.18(C2×C4), (C2×D44).1C2, (C2×C22).15D4, (C2×C4).71D22, C11⋊2(D4⋊C4), C2.8(D22⋊C4), C2.3(C8⋊D11), C4.20(C11⋊D4), C22.7(C22⋊C4), (C2×C44).83C22, SmallGroup(352,27)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2.D88
G = < a,b,c | a2=b88=1, c2=a, ab=ba, ac=ca, cbc-1=ab-1 >
44C2
44C2
22C22
22C22
44C22
44C22
44C4
4D11
4D11
2C8
11D4
11D4
22C2×C4
22C23
22D4
2D22
2D22
4D22
4D22
11C4⋊C4
11C2×D4
2C88
2D44
11D4⋊C4
Smallest permutation representation of C2.D88
►On 176 points
Generators in S176
(1 95)(2 96)(3 97)(4 98)(5 99)(6 100)(7 101)(8 102)(9 103)(10 104)(11 105)(12 106)(13 107)(14 108)(15 109)(16 110)(17 111)(18 112)(19 113)(20 114)(21 115)(22 116)(23 117)(24 118)(25 119)(26 120)(27 121)(28 122)(29 123)(30 124)(31 125)(32 126)(33 127)(34 128)(35 129)(36 130)(37 131)(38 132)(39 133)(40 134)(41 135)(42 136)(43 137)(44 138)(45 139)(46 140)(47 141)(48 142)(49 143)(50 144)(51 145)(52 146)(53 147)(54 148)(55 149)(56 150)(57 151)(58 152)(59 153)(60 154)(61 155)(62 156)(63 157)(64 158)(65 159)(66 160)(67 161)(68 162)(69 163)(70 164)(71 165)(72 166)(73 167)(74 168)(75 169)(76 170)(77 171)(78 172)(79 173)(80 174)(81 175)(82 176)(83 89)(84 90)(85 91)(86 92)(87 93)(88 94)
(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 94 95 88)(2 87 96 93)(3 92 97 86)(4 85 98 91)(5 90 99 84)(6 83 100 89)(7 176 101 82)(8 81 102 175)(9 174 103 80)(10 79 104 173)(11 172 105 78)(12 77 106 171)(13 170 107 76)(14 75 108 169)(15 168 109 74)(16 73 110 167)(17 166 111 72)(18 71 112 165)(19 164 113 70)(20 69 114 163)(21 162 115 68)(22 67 116 161)(23 160 117 66)(24 65 118 159)(25 158 119 64)(26 63 120 157)(27 156 121 62)(28 61 122 155)(29 154 123 60)(30 59 124 153)(31 152 125 58)(32 57 126 151)(33 150 127 56)(34 55 128 149)(35 148 129 54)(36 53 130 147)(37 146 131 52)(38 51 132 145)(39 144 133 50)(40 49 134 143)(41 142 135 48)(42 47 136 141)(43 140 137 46)(44 45 138 139)
G:=sub<Sym(176)| (1,95)(2,96)(3,97)(4,98)(5,99)(6,100)(7,101)(8,102)(9,103)(10,104)(11,105)(12,106)(13,107)(14,108)(15,109)(16,110)(17,111)(18,112)(19,113)(20,114)(21,115)(22,116)(23,117)(24,118)(25,119)(26,120)(27,121)(28,122)(29,123)(30,124)(31,125)(32,126)(33,127)(34,128)(35,129)(36,130)(37,131)(38,132)(39,133)(40,134)(41,135)(42,136)(43,137)(44,138)(45,139)(46,140)(47,141)(48,142)(49,143)(50,144)(51,145)(52,146)(53,147)(54,148)(55,149)(56,150)(57,151)(58,152)(59,153)(60,154)(61,155)(62,156)(63,157)(64,158)(65,159)(66,160)(67,161)(68,162)(69,163)(70,164)(71,165)(72,166)(73,167)(74,168)(75,169)(76,170)(77,171)(78,172)(79,173)(80,174)(81,175)(82,176)(83,89)(84,90)(85,91)(86,92)(87,93)(88,94), (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), (1,94,95,88)(2,87,96,93)(3,92,97,86)(4,85,98,91)(5,90,99,84)(6,83,100,89)(7,176,101,82)(8,81,102,175)(9,174,103,80)(10,79,104,173)(11,172,105,78)(12,77,106,171)(13,170,107,76)(14,75,108,169)(15,168,109,74)(16,73,110,167)(17,166,111,72)(18,71,112,165)(19,164,113,70)(20,69,114,163)(21,162,115,68)(22,67,116,161)(23,160,117,66)(24,65,118,159)(25,158,119,64)(26,63,120,157)(27,156,121,62)(28,61,122,155)(29,154,123,60)(30,59,124,153)(31,152,125,58)(32,57,126,151)(33,150,127,56)(34,55,128,149)(35,148,129,54)(36,53,130,147)(37,146,131,52)(38,51,132,145)(39,144,133,50)(40,49,134,143)(41,142,135,48)(42,47,136,141)(43,140,137,46)(44,45,138,139)>;
G:=Group( (1,95)(2,96)(3,97)(4,98)(5,99)(6,100)(7,101)(8,102)(9,103)(10,104)(11,105)(12,106)(13,107)(14,108)(15,109)(16,110)(17,111)(18,112)(19,113)(20,114)(21,115)(22,116)(23,117)(24,118)(25,119)(26,120)(27,121)(28,122)(29,123)(30,124)(31,125)(32,126)(33,127)(34,128)(35,129)(36,130)(37,131)(38,132)(39,133)(40,134)(41,135)(42,136)(43,137)(44,138)(45,139)(46,140)(47,141)(48,142)(49,143)(50,144)(51,145)(52,146)(53,147)(54,148)(55,149)(56,150)(57,151)(58,152)(59,153)(60,154)(61,155)(62,156)(63,157)(64,158)(65,159)(66,160)(67,161)(68,162)(69,163)(70,164)(71,165)(72,166)(73,167)(74,168)(75,169)(76,170)(77,171)(78,172)(79,173)(80,174)(81,175)(82,176)(83,89)(84,90)(85,91)(86,92)(87,93)(88,94), (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), (1,94,95,88)(2,87,96,93)(3,92,97,86)(4,85,98,91)(5,90,99,84)(6,83,100,89)(7,176,101,82)(8,81,102,175)(9,174,103,80)(10,79,104,173)(11,172,105,78)(12,77,106,171)(13,170,107,76)(14,75,108,169)(15,168,109,74)(16,73,110,167)(17,166,111,72)(18,71,112,165)(19,164,113,70)(20,69,114,163)(21,162,115,68)(22,67,116,161)(23,160,117,66)(24,65,118,159)(25,158,119,64)(26,63,120,157)(27,156,121,62)(28,61,122,155)(29,154,123,60)(30,59,124,153)(31,152,125,58)(32,57,126,151)(33,150,127,56)(34,55,128,149)(35,148,129,54)(36,53,130,147)(37,146,131,52)(38,51,132,145)(39,144,133,50)(40,49,134,143)(41,142,135,48)(42,47,136,141)(43,140,137,46)(44,45,138,139) );
G=PermutationGroup([[(1,95),(2,96),(3,97),(4,98),(5,99),(6,100),(7,101),(8,102),(9,103),(10,104),(11,105),(12,106),(13,107),(14,108),(15,109),(16,110),(17,111),(18,112),(19,113),(20,114),(21,115),(22,116),(23,117),(24,118),(25,119),(26,120),(27,121),(28,122),(29,123),(30,124),(31,125),(32,126),(33,127),(34,128),(35,129),(36,130),(37,131),(38,132),(39,133),(40,134),(41,135),(42,136),(43,137),(44,138),(45,139),(46,140),(47,141),(48,142),(49,143),(50,144),(51,145),(52,146),(53,147),(54,148),(55,149),(56,150),(57,151),(58,152),(59,153),(60,154),(61,155),(62,156),(63,157),(64,158),(65,159),(66,160),(67,161),(68,162),(69,163),(70,164),(71,165),(72,166),(73,167),(74,168),(75,169),(76,170),(77,171),(78,172),(79,173),(80,174),(81,175),(82,176),(83,89),(84,90),(85,91),(86,92),(87,93),(88,94)], [(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)], [(1,94,95,88),(2,87,96,93),(3,92,97,86),(4,85,98,91),(5,90,99,84),(6,83,100,89),(7,176,101,82),(8,81,102,175),(9,174,103,80),(10,79,104,173),(11,172,105,78),(12,77,106,171),(13,170,107,76),(14,75,108,169),(15,168,109,74),(16,73,110,167),(17,166,111,72),(18,71,112,165),(19,164,113,70),(20,69,114,163),(21,162,115,68),(22,67,116,161),(23,160,117,66),(24,65,118,159),(25,158,119,64),(26,63,120,157),(27,156,121,62),(28,61,122,155),(29,154,123,60),(30,59,124,153),(31,152,125,58),(32,57,126,151),(33,150,127,56),(34,55,128,149),(35,148,129,54),(36,53,130,147),(37,146,131,52),(38,51,132,145),(39,144,133,50),(40,49,134,143),(41,142,135,48),(42,47,136,141),(43,140,137,46),(44,45,138,139)]])
94 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 8A | 8B | 8C | 8D | 11A | ··· | 11E | 22A | ··· | 22O | 44A | ··· | 44T | 88A | ··· | 88AN |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 1 | 1 | 44 | 44 | 2 | 2 | 44 | 44 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
94 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D4 | D8 | SD16 | D11 | D22 | C4×D11 | C11⋊D4 | D44 | C8⋊D11 | D88 |
| kernel | C2.D88 | C44⋊C4 | C2×C88 | C2×D44 | D44 | C44 | C2×C22 | C22 | C22 | C2×C8 | C2×C4 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 5 | 5 | 10 | 10 | 10 | 20 | 20 |
Matrix representation of C2.D88 ►in GL4(𝔽89) generated by
| 88 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 88 | 0 |
| 0 | 0 | 0 | 88 |
| 11 | 39 | 0 | 0 |
| 63 | 21 | 0 | 0 |
| 0 | 0 | 15 | 77 |
| 0 | 0 | 12 | 20 |
| 85 | 75 | 0 | 0 |
| 33 | 4 | 0 | 0 |
| 0 | 0 | 15 | 77 |
| 0 | 0 | 4 | 74 |
G:=sub<GL(4,GF(89))| [88,0,0,0,0,88,0,0,0,0,88,0,0,0,0,88],[11,63,0,0,39,21,0,0,0,0,15,12,0,0,77,20],[85,33,0,0,75,4,0,0,0,0,15,4,0,0,77,74] >;
C2.D88 in GAP, Magma, Sage, TeX
C_2.D_{88} % in TeX
G:=Group("C2.D88"); // GroupNames label
G:=SmallGroup(352,27);
// by ID
G=gap.SmallGroup(352,27);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,73,79,362,86,11525]);
// Polycyclic
G:=Group<a,b,c|a^2=b^88=1,c^2=a,a*b=b*a,a*c=c*a,c*b*c^-1=a*b^-1>;
// generators/relations
Export