direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D88, C22⋊1D8, C8⋊7D22, C4.7D44, C88⋊8C22, C44.30D4, D44⋊3C22, C44.29C23, C22.13D44, C11⋊1(C2×D8), (C2×C88)⋊5C2, (C2×C8)⋊3D11, (C2×D44)⋊5C2, C2.12(C2×D44), C22.10(C2×D4), (C2×C22).17D4, (C2×C4).80D22, (C2×C44).89C22, C4.27(C22×D11), SmallGroup(352,98)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D88
G = < a,b,c | a2=b88=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 730 in 76 conjugacy classes, 33 normal (15 characteristic)
C1, C2, C2, C2, C4, C22, C22, C8, C2×C4, D4, C23, C11, C2×C8, D8, C2×D4, D11, C22, C22, C2×D8, C44, D22, C2×C22, C88, D44, D44, C2×C44, C22×D11, D88, C2×C88, C2×D44, C2×D88
Quotients: C1, C2, C22, D4, C23, D8, C2×D4, D11, C2×D8, D22, D44, C22×D11, D88, C2×D44, C2×D88
►On 176 points
Generators in S176
(1 155)(2 156)(3 157)(4 158)(5 159)(6 160)(7 161)(8 162)(9 163)(10 164)(11 165)(12 166)(13 167)(14 168)(15 169)(16 170)(17 171)(18 172)(19 173)(20 174)(21 175)(22 176)(23 89)(24 90)(25 91)(26 92)(27 93)(28 94)(29 95)(30 96)(31 97)(32 98)(33 99)(34 100)(35 101)(36 102)(37 103)(38 104)(39 105)(40 106)(41 107)(42 108)(43 109)(44 110)(45 111)(46 112)(47 113)(48 114)(49 115)(50 116)(51 117)(52 118)(53 119)(54 120)(55 121)(56 122)(57 123)(58 124)(59 125)(60 126)(61 127)(62 128)(63 129)(64 130)(65 131)(66 132)(67 133)(68 134)(69 135)(70 136)(71 137)(72 138)(73 139)(74 140)(75 141)(76 142)(77 143)(78 144)(79 145)(80 146)(81 147)(82 148)(83 149)(84 150)(85 151)(86 152)(87 153)(88 154)
(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 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)(34 88)(35 87)(36 86)(37 85)(38 84)(39 83)(40 82)(41 81)(42 80)(43 79)(44 78)(45 77)(46 76)(47 75)(48 74)(49 73)(50 72)(51 71)(52 70)(53 69)(54 68)(55 67)(56 66)(57 65)(58 64)(59 63)(60 62)(89 165)(90 164)(91 163)(92 162)(93 161)(94 160)(95 159)(96 158)(97 157)(98 156)(99 155)(100 154)(101 153)(102 152)(103 151)(104 150)(105 149)(106 148)(107 147)(108 146)(109 145)(110 144)(111 143)(112 142)(113 141)(114 140)(115 139)(116 138)(117 137)(118 136)(119 135)(120 134)(121 133)(122 132)(123 131)(124 130)(125 129)(126 128)(166 176)(167 175)(168 174)(169 173)(170 172)
G:=sub<Sym(176)| (1,155)(2,156)(3,157)(4,158)(5,159)(6,160)(7,161)(8,162)(9,163)(10,164)(11,165)(12,166)(13,167)(14,168)(15,169)(16,170)(17,171)(18,172)(19,173)(20,174)(21,175)(22,176)(23,89)(24,90)(25,91)(26,92)(27,93)(28,94)(29,95)(30,96)(31,97)(32,98)(33,99)(34,100)(35,101)(36,102)(37,103)(38,104)(39,105)(40,106)(41,107)(42,108)(43,109)(44,110)(45,111)(46,112)(47,113)(48,114)(49,115)(50,116)(51,117)(52,118)(53,119)(54,120)(55,121)(56,122)(57,123)(58,124)(59,125)(60,126)(61,127)(62,128)(63,129)(64,130)(65,131)(66,132)(67,133)(68,134)(69,135)(70,136)(71,137)(72,138)(73,139)(74,140)(75,141)(76,142)(77,143)(78,144)(79,145)(80,146)(81,147)(82,148)(83,149)(84,150)(85,151)(86,152)(87,153)(88,154), (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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(34,88)(35,87)(36,86)(37,85)(38,84)(39,83)(40,82)(41,81)(42,80)(43,79)(44,78)(45,77)(46,76)(47,75)(48,74)(49,73)(50,72)(51,71)(52,70)(53,69)(54,68)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62)(89,165)(90,164)(91,163)(92,162)(93,161)(94,160)(95,159)(96,158)(97,157)(98,156)(99,155)(100,154)(101,153)(102,152)(103,151)(104,150)(105,149)(106,148)(107,147)(108,146)(109,145)(110,144)(111,143)(112,142)(113,141)(114,140)(115,139)(116,138)(117,137)(118,136)(119,135)(120,134)(121,133)(122,132)(123,131)(124,130)(125,129)(126,128)(166,176)(167,175)(168,174)(169,173)(170,172)>;
G:=Group( (1,155)(2,156)(3,157)(4,158)(5,159)(6,160)(7,161)(8,162)(9,163)(10,164)(11,165)(12,166)(13,167)(14,168)(15,169)(16,170)(17,171)(18,172)(19,173)(20,174)(21,175)(22,176)(23,89)(24,90)(25,91)(26,92)(27,93)(28,94)(29,95)(30,96)(31,97)(32,98)(33,99)(34,100)(35,101)(36,102)(37,103)(38,104)(39,105)(40,106)(41,107)(42,108)(43,109)(44,110)(45,111)(46,112)(47,113)(48,114)(49,115)(50,116)(51,117)(52,118)(53,119)(54,120)(55,121)(56,122)(57,123)(58,124)(59,125)(60,126)(61,127)(62,128)(63,129)(64,130)(65,131)(66,132)(67,133)(68,134)(69,135)(70,136)(71,137)(72,138)(73,139)(74,140)(75,141)(76,142)(77,143)(78,144)(79,145)(80,146)(81,147)(82,148)(83,149)(84,150)(85,151)(86,152)(87,153)(88,154), (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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,24)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)(34,88)(35,87)(36,86)(37,85)(38,84)(39,83)(40,82)(41,81)(42,80)(43,79)(44,78)(45,77)(46,76)(47,75)(48,74)(49,73)(50,72)(51,71)(52,70)(53,69)(54,68)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62)(89,165)(90,164)(91,163)(92,162)(93,161)(94,160)(95,159)(96,158)(97,157)(98,156)(99,155)(100,154)(101,153)(102,152)(103,151)(104,150)(105,149)(106,148)(107,147)(108,146)(109,145)(110,144)(111,143)(112,142)(113,141)(114,140)(115,139)(116,138)(117,137)(118,136)(119,135)(120,134)(121,133)(122,132)(123,131)(124,130)(125,129)(126,128)(166,176)(167,175)(168,174)(169,173)(170,172) );
G=PermutationGroup([[(1,155),(2,156),(3,157),(4,158),(5,159),(6,160),(7,161),(8,162),(9,163),(10,164),(11,165),(12,166),(13,167),(14,168),(15,169),(16,170),(17,171),(18,172),(19,173),(20,174),(21,175),(22,176),(23,89),(24,90),(25,91),(26,92),(27,93),(28,94),(29,95),(30,96),(31,97),(32,98),(33,99),(34,100),(35,101),(36,102),(37,103),(38,104),(39,105),(40,106),(41,107),(42,108),(43,109),(44,110),(45,111),(46,112),(47,113),(48,114),(49,115),(50,116),(51,117),(52,118),(53,119),(54,120),(55,121),(56,122),(57,123),(58,124),(59,125),(60,126),(61,127),(62,128),(63,129),(64,130),(65,131),(66,132),(67,133),(68,134),(69,135),(70,136),(71,137),(72,138),(73,139),(74,140),(75,141),(76,142),(77,143),(78,144),(79,145),(80,146),(81,147),(82,148),(83,149),(84,150),(85,151),(86,152),(87,153),(88,154)], [(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)], [(1,33),(2,32),(3,31),(4,30),(5,29),(6,28),(7,27),(8,26),(9,25),(10,24),(11,23),(12,22),(13,21),(14,20),(15,19),(16,18),(34,88),(35,87),(36,86),(37,85),(38,84),(39,83),(40,82),(41,81),(42,80),(43,79),(44,78),(45,77),(46,76),(47,75),(48,74),(49,73),(50,72),(51,71),(52,70),(53,69),(54,68),(55,67),(56,66),(57,65),(58,64),(59,63),(60,62),(89,165),(90,164),(91,163),(92,162),(93,161),(94,160),(95,159),(96,158),(97,157),(98,156),(99,155),(100,154),(101,153),(102,152),(103,151),(104,150),(105,149),(106,148),(107,147),(108,146),(109,145),(110,144),(111,143),(112,142),(113,141),(114,140),(115,139),(116,138),(117,137),(118,136),(119,135),(120,134),(121,133),(122,132),(123,131),(124,130),(125,129),(126,128),(166,176),(167,175),(168,174),(169,173),(170,172)]])
94 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 8A | 8B | 8C | 8D | 11A | ··· | 11E | 22A | ··· | 22O | 44A | ··· | 44T | 88A | ··· | 88AN |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 | 88 | ··· | 88 |
| size | 1 | 1 | 1 | 1 | 44 | 44 | 44 | 44 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | C2 | D4 | D4 | D8 | D11 | D22 | D22 | D44 | D44 | D88 |
| kernel | C2×D88 | D88 | C2×C88 | C2×D44 | C44 | C2×C22 | C22 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 1 | 1 | 4 | 5 | 10 | 5 | 10 | 10 | 40 |
Matrix representation of C2×D88 ►in GL3(𝔽89) generated by
| 88 | 0 | 0 |
| 0 | 88 | 0 |
| 0 | 0 | 88 |
| 1 | 0 | 0 |
| 0 | 6 | 22 |
| 0 | 1 | 78 |
| 88 | 0 | 0 |
| 0 | 84 | 47 |
| 0 | 26 | 5 |
G:=sub<GL(3,GF(89))| [88,0,0,0,88,0,0,0,88],[1,0,0,0,6,1,0,22,78],[88,0,0,0,84,26,0,47,5] >;
C2×D88 in GAP, Magma, Sage, TeX
C_2\times D_{88} % in TeX
G:=Group("C2xD88"); // GroupNames label
G:=SmallGroup(352,98);
// by ID
G=gap.SmallGroup(352,98);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,218,122,579,69,11525]);
// 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^-1>;
// generators/relations