Copied to
clipboard

## G = C2×C88⋊C2order 352 = 25·11

### Direct product of C2 and C88⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C2×C88⋊C2
 Chief series C1 — C11 — C22 — C44 — C4×D11 — C2×C4×D11 — C2×C88⋊C2
 Lower central C11 — C22 — C2×C88⋊C2
 Upper central C1 — C2×C4 — C2×C8

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

Smallest permutation representation of 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

׿
×
𝔽