direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C88⋊C2, C8⋊9D22, C88⋊11C22, C22⋊1M4(2), C44.36C23, (C2×C88)⋊9C2, (C2×C8)⋊6D11, C11⋊C8⋊10C22, C44.27(C2×C4), D22.5(C2×C4), (C4×D11).3C4, C4.24(C4×D11), (C2×C4).98D22, C11⋊1(C2×M4(2)), C22.13(C22×C4), (C2×Dic11).5C4, Dic11.6(C2×C4), (C22×D11).3C4, C22.14(C4×D11), C4.36(C22×D11), (C2×C44).111C22, (C4×D11).14C22, (C2×C11⋊C8)⋊11C2, C2.14(C2×C4×D11), (C2×C4×D11).10C2, (C2×C22).15(C2×C4), SmallGroup(352,95)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C88⋊C2
G = < a,b,c | a2=b88=c2=1, ab=ba, ac=ca, cbc=b21 >
Subgroups: 346 in 68 conjugacy classes, 41 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, C23, C11, C2×C8, C2×C8, M4(2), C22×C4, D11, C22, C22, C2×M4(2), Dic11, C44, D22, D22, C2×C22, C11⋊C8, C88, C4×D11, C2×Dic11, C2×C44, C22×D11, C88⋊C2, C2×C11⋊C8, C2×C88, C2×C4×D11, C2×C88⋊C2
Quotients: C1, C2, C4, C22, C2×C4, C23, M4(2), C22×C4, D11, C2×M4(2), D22, C4×D11, C22×D11, C88⋊C2, C2×C4×D11, C2×C88⋊C2
►On 176 points
Generators in S176
(1 93)(2 94)(3 95)(4 96)(5 97)(6 98)(7 99)(8 100)(9 101)(10 102)(11 103)(12 104)(13 105)(14 106)(15 107)(16 108)(17 109)(18 110)(19 111)(20 112)(21 113)(22 114)(23 115)(24 116)(25 117)(26 118)(27 119)(28 120)(29 121)(30 122)(31 123)(32 124)(33 125)(34 126)(35 127)(36 128)(37 129)(38 130)(39 131)(40 132)(41 133)(42 134)(43 135)(44 136)(45 137)(46 138)(47 139)(48 140)(49 141)(50 142)(51 143)(52 144)(53 145)(54 146)(55 147)(56 148)(57 149)(58 150)(59 151)(60 152)(61 153)(62 154)(63 155)(64 156)(65 157)(66 158)(67 159)(68 160)(69 161)(70 162)(71 163)(72 164)(73 165)(74 166)(75 167)(76 168)(77 169)(78 170)(79 171)(80 172)(81 173)(82 174)(83 175)(84 176)(85 89)(86 90)(87 91)(88 92)
(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)
(2 22)(3 43)(4 64)(5 85)(6 18)(7 39)(8 60)(9 81)(10 14)(11 35)(12 56)(13 77)(15 31)(16 52)(17 73)(19 27)(20 48)(21 69)(24 44)(25 65)(26 86)(28 40)(29 61)(30 82)(32 36)(33 57)(34 78)(37 53)(38 74)(41 49)(42 70)(46 66)(47 87)(50 62)(51 83)(54 58)(55 79)(59 75)(63 71)(68 88)(72 84)(76 80)(89 97)(90 118)(91 139)(92 160)(94 114)(95 135)(96 156)(98 110)(99 131)(100 152)(101 173)(102 106)(103 127)(104 148)(105 169)(107 123)(108 144)(109 165)(111 119)(112 140)(113 161)(116 136)(117 157)(120 132)(121 153)(122 174)(124 128)(125 149)(126 170)(129 145)(130 166)(133 141)(134 162)(138 158)(142 154)(143 175)(146 150)(147 171)(151 167)(155 163)(164 176)(168 172)
G:=sub<Sym(176)| (1,93)(2,94)(3,95)(4,96)(5,97)(6,98)(7,99)(8,100)(9,101)(10,102)(11,103)(12,104)(13,105)(14,106)(15,107)(16,108)(17,109)(18,110)(19,111)(20,112)(21,113)(22,114)(23,115)(24,116)(25,117)(26,118)(27,119)(28,120)(29,121)(30,122)(31,123)(32,124)(33,125)(34,126)(35,127)(36,128)(37,129)(38,130)(39,131)(40,132)(41,133)(42,134)(43,135)(44,136)(45,137)(46,138)(47,139)(48,140)(49,141)(50,142)(51,143)(52,144)(53,145)(54,146)(55,147)(56,148)(57,149)(58,150)(59,151)(60,152)(61,153)(62,154)(63,155)(64,156)(65,157)(66,158)(67,159)(68,160)(69,161)(70,162)(71,163)(72,164)(73,165)(74,166)(75,167)(76,168)(77,169)(78,170)(79,171)(80,172)(81,173)(82,174)(83,175)(84,176)(85,89)(86,90)(87,91)(88,92), (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), (2,22)(3,43)(4,64)(5,85)(6,18)(7,39)(8,60)(9,81)(10,14)(11,35)(12,56)(13,77)(15,31)(16,52)(17,73)(19,27)(20,48)(21,69)(24,44)(25,65)(26,86)(28,40)(29,61)(30,82)(32,36)(33,57)(34,78)(37,53)(38,74)(41,49)(42,70)(46,66)(47,87)(50,62)(51,83)(54,58)(55,79)(59,75)(63,71)(68,88)(72,84)(76,80)(89,97)(90,118)(91,139)(92,160)(94,114)(95,135)(96,156)(98,110)(99,131)(100,152)(101,173)(102,106)(103,127)(104,148)(105,169)(107,123)(108,144)(109,165)(111,119)(112,140)(113,161)(116,136)(117,157)(120,132)(121,153)(122,174)(124,128)(125,149)(126,170)(129,145)(130,166)(133,141)(134,162)(138,158)(142,154)(143,175)(146,150)(147,171)(151,167)(155,163)(164,176)(168,172)>;
G:=Group( (1,93)(2,94)(3,95)(4,96)(5,97)(6,98)(7,99)(8,100)(9,101)(10,102)(11,103)(12,104)(13,105)(14,106)(15,107)(16,108)(17,109)(18,110)(19,111)(20,112)(21,113)(22,114)(23,115)(24,116)(25,117)(26,118)(27,119)(28,120)(29,121)(30,122)(31,123)(32,124)(33,125)(34,126)(35,127)(36,128)(37,129)(38,130)(39,131)(40,132)(41,133)(42,134)(43,135)(44,136)(45,137)(46,138)(47,139)(48,140)(49,141)(50,142)(51,143)(52,144)(53,145)(54,146)(55,147)(56,148)(57,149)(58,150)(59,151)(60,152)(61,153)(62,154)(63,155)(64,156)(65,157)(66,158)(67,159)(68,160)(69,161)(70,162)(71,163)(72,164)(73,165)(74,166)(75,167)(76,168)(77,169)(78,170)(79,171)(80,172)(81,173)(82,174)(83,175)(84,176)(85,89)(86,90)(87,91)(88,92), (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), (2,22)(3,43)(4,64)(5,85)(6,18)(7,39)(8,60)(9,81)(10,14)(11,35)(12,56)(13,77)(15,31)(16,52)(17,73)(19,27)(20,48)(21,69)(24,44)(25,65)(26,86)(28,40)(29,61)(30,82)(32,36)(33,57)(34,78)(37,53)(38,74)(41,49)(42,70)(46,66)(47,87)(50,62)(51,83)(54,58)(55,79)(59,75)(63,71)(68,88)(72,84)(76,80)(89,97)(90,118)(91,139)(92,160)(94,114)(95,135)(96,156)(98,110)(99,131)(100,152)(101,173)(102,106)(103,127)(104,148)(105,169)(107,123)(108,144)(109,165)(111,119)(112,140)(113,161)(116,136)(117,157)(120,132)(121,153)(122,174)(124,128)(125,149)(126,170)(129,145)(130,166)(133,141)(134,162)(138,158)(142,154)(143,175)(146,150)(147,171)(151,167)(155,163)(164,176)(168,172) );
G=PermutationGroup([[(1,93),(2,94),(3,95),(4,96),(5,97),(6,98),(7,99),(8,100),(9,101),(10,102),(11,103),(12,104),(13,105),(14,106),(15,107),(16,108),(17,109),(18,110),(19,111),(20,112),(21,113),(22,114),(23,115),(24,116),(25,117),(26,118),(27,119),(28,120),(29,121),(30,122),(31,123),(32,124),(33,125),(34,126),(35,127),(36,128),(37,129),(38,130),(39,131),(40,132),(41,133),(42,134),(43,135),(44,136),(45,137),(46,138),(47,139),(48,140),(49,141),(50,142),(51,143),(52,144),(53,145),(54,146),(55,147),(56,148),(57,149),(58,150),(59,151),(60,152),(61,153),(62,154),(63,155),(64,156),(65,157),(66,158),(67,159),(68,160),(69,161),(70,162),(71,163),(72,164),(73,165),(74,166),(75,167),(76,168),(77,169),(78,170),(79,171),(80,172),(81,173),(82,174),(83,175),(84,176),(85,89),(86,90),(87,91),(88,92)], [(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)], [(2,22),(3,43),(4,64),(5,85),(6,18),(7,39),(8,60),(9,81),(10,14),(11,35),(12,56),(13,77),(15,31),(16,52),(17,73),(19,27),(20,48),(21,69),(24,44),(25,65),(26,86),(28,40),(29,61),(30,82),(32,36),(33,57),(34,78),(37,53),(38,74),(41,49),(42,70),(46,66),(47,87),(50,62),(51,83),(54,58),(55,79),(59,75),(63,71),(68,88),(72,84),(76,80),(89,97),(90,118),(91,139),(92,160),(94,114),(95,135),(96,156),(98,110),(99,131),(100,152),(101,173),(102,106),(103,127),(104,148),(105,169),(107,123),(108,144),(109,165),(111,119),(112,140),(113,161),(116,136),(117,157),(120,132),(121,153),(122,174),(124,128),(125,149),(126,170),(129,145),(130,166),(133,141),(134,162),(138,158),(142,154),(143,175),(146,150),(147,171),(151,167),(155,163),(164,176),(168,172)]])
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 11A | ··· | 11E | 22A | ··· | 22O | 44A | ··· | 44T | 88A | ··· | 88AN |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 1 | 1 | 22 | 22 | 1 | 1 | 1 | 1 | 22 | 22 | 2 | 2 | 2 | 2 | 22 | 22 | 22 | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | M4(2) | D11 | D22 | D22 | C4×D11 | C4×D11 | C88⋊C2 |
| kernel | C2×C88⋊C2 | C88⋊C2 | C2×C11⋊C8 | C2×C88 | C2×C4×D11 | C4×D11 | C2×Dic11 | C22×D11 | C22 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 5 | 10 | 5 | 10 | 10 | 40 |
Matrix representation of C2×C88⋊C2 ►in GL3(𝔽89) generated by
| 88 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 56 | 84 |
| 0 | 5 | 24 |
| 88 | 0 | 0 |
| 0 | 1 | 47 |
| 0 | 0 | 88 |
G:=sub<GL(3,GF(89))| [88,0,0,0,1,0,0,0,1],[1,0,0,0,56,5,0,84,24],[88,0,0,0,1,0,0,47,88] >;
C2×C88⋊C2 in GAP, Magma, Sage, TeX
C_2\times C_{88}\rtimes C_2 % in TeX
G:=Group("C2xC88:C2"); // GroupNames label
G:=SmallGroup(352,95);
// by ID
G=gap.SmallGroup(352,95);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,362,50,69,11525]);
// Polycyclic
G:=Group<a,b,c|a^2=b^88=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^21>;
// generators/relations