Copied to
clipboard

## G = C2×C8⋊D11order 352 = 25·11

### Direct product of C2 and C8⋊D11

Series: Derived Chief Lower central Upper central

 Derived series C1 — C44 — C2×C8⋊D11
 Chief series C1 — C11 — C22 — C44 — D44 — C2×D44 — C2×C8⋊D11
 Lower central C11 — C22 — C44 — C2×C8⋊D11
 Upper central C1 — C22 — C2×C4 — C2×C8

Generators and relations for C2×C8⋊D11
G = < a,b,c,d | a2=b8=c11=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b3, dcd=c-1 >

Subgroups: 538 in 68 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C11, C2×C8, SD16, C2×D4, C2×Q8, D11, C22, C22, C2×SD16, Dic11, C44, D22, C2×C22, C88, Dic22, Dic22, D44, D44, C2×Dic11, C2×C44, C22×D11, C8⋊D11, C2×C88, C2×Dic22, C2×D44, C2×C8⋊D11
Quotients: C1, C2, C22, D4, C23, SD16, C2×D4, D11, C2×SD16, D22, D44, C22×D11, C8⋊D11, C2×D44, C2×C8⋊D11

Smallest permutation representation of C2×C8⋊D11
On 176 points
Generators in S176
(1 89)(2 90)(3 91)(4 92)(5 93)(6 94)(7 95)(8 96)(9 97)(10 98)(11 99)(12 100)(13 101)(14 102)(15 103)(16 104)(17 105)(18 106)(19 107)(20 108)(21 109)(22 110)(23 111)(24 112)(25 113)(26 114)(27 115)(28 116)(29 117)(30 118)(31 119)(32 120)(33 121)(34 122)(35 123)(36 124)(37 125)(38 126)(39 127)(40 128)(41 129)(42 130)(43 131)(44 132)(45 133)(46 134)(47 135)(48 136)(49 137)(50 138)(51 139)(52 140)(53 141)(54 142)(55 143)(56 144)(57 145)(58 146)(59 147)(60 148)(61 149)(62 150)(63 151)(64 152)(65 153)(66 154)(67 155)(68 156)(69 157)(70 158)(71 159)(72 160)(73 161)(74 162)(75 163)(76 164)(77 165)(78 166)(79 167)(80 168)(81 169)(82 170)(83 171)(84 172)(85 173)(86 174)(87 175)(88 176)
(1 166 34 144 12 155 23 133)(2 167 35 145 13 156 24 134)(3 168 36 146 14 157 25 135)(4 169 37 147 15 158 26 136)(5 170 38 148 16 159 27 137)(6 171 39 149 17 160 28 138)(7 172 40 150 18 161 29 139)(8 173 41 151 19 162 30 140)(9 174 42 152 20 163 31 141)(10 175 43 153 21 164 32 142)(11 176 44 154 22 165 33 143)(45 89 78 122 56 100 67 111)(46 90 79 123 57 101 68 112)(47 91 80 124 58 102 69 113)(48 92 81 125 59 103 70 114)(49 93 82 126 60 104 71 115)(50 94 83 127 61 105 72 116)(51 95 84 128 62 106 73 117)(52 96 85 129 63 107 74 118)(53 97 86 130 64 108 75 119)(54 98 87 131 65 109 76 120)(55 99 88 132 66 110 77 121)
(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 99)(2 98)(3 97)(4 96)(5 95)(6 94)(7 93)(8 92)(9 91)(10 90)(11 89)(12 110)(13 109)(14 108)(15 107)(16 106)(17 105)(18 104)(19 103)(20 102)(21 101)(22 100)(23 132)(24 131)(25 130)(26 129)(27 128)(28 127)(29 126)(30 125)(31 124)(32 123)(33 122)(34 121)(35 120)(36 119)(37 118)(38 117)(39 116)(40 115)(41 114)(42 113)(43 112)(44 111)(45 165)(46 164)(47 163)(48 162)(49 161)(50 160)(51 159)(52 158)(53 157)(54 156)(55 155)(56 176)(57 175)(58 174)(59 173)(60 172)(61 171)(62 170)(63 169)(64 168)(65 167)(66 166)(67 143)(68 142)(69 141)(70 140)(71 139)(72 138)(73 137)(74 136)(75 135)(76 134)(77 133)(78 154)(79 153)(80 152)(81 151)(82 150)(83 149)(84 148)(85 147)(86 146)(87 145)(88 144)

G:=sub<Sym(176)| (1,89)(2,90)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,97)(10,98)(11,99)(12,100)(13,101)(14,102)(15,103)(16,104)(17,105)(18,106)(19,107)(20,108)(21,109)(22,110)(23,111)(24,112)(25,113)(26,114)(27,115)(28,116)(29,117)(30,118)(31,119)(32,120)(33,121)(34,122)(35,123)(36,124)(37,125)(38,126)(39,127)(40,128)(41,129)(42,130)(43,131)(44,132)(45,133)(46,134)(47,135)(48,136)(49,137)(50,138)(51,139)(52,140)(53,141)(54,142)(55,143)(56,144)(57,145)(58,146)(59,147)(60,148)(61,149)(62,150)(63,151)(64,152)(65,153)(66,154)(67,155)(68,156)(69,157)(70,158)(71,159)(72,160)(73,161)(74,162)(75,163)(76,164)(77,165)(78,166)(79,167)(80,168)(81,169)(82,170)(83,171)(84,172)(85,173)(86,174)(87,175)(88,176), (1,166,34,144,12,155,23,133)(2,167,35,145,13,156,24,134)(3,168,36,146,14,157,25,135)(4,169,37,147,15,158,26,136)(5,170,38,148,16,159,27,137)(6,171,39,149,17,160,28,138)(7,172,40,150,18,161,29,139)(8,173,41,151,19,162,30,140)(9,174,42,152,20,163,31,141)(10,175,43,153,21,164,32,142)(11,176,44,154,22,165,33,143)(45,89,78,122,56,100,67,111)(46,90,79,123,57,101,68,112)(47,91,80,124,58,102,69,113)(48,92,81,125,59,103,70,114)(49,93,82,126,60,104,71,115)(50,94,83,127,61,105,72,116)(51,95,84,128,62,106,73,117)(52,96,85,129,63,107,74,118)(53,97,86,130,64,108,75,119)(54,98,87,131,65,109,76,120)(55,99,88,132,66,110,77,121), (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), (1,99)(2,98)(3,97)(4,96)(5,95)(6,94)(7,93)(8,92)(9,91)(10,90)(11,89)(12,110)(13,109)(14,108)(15,107)(16,106)(17,105)(18,104)(19,103)(20,102)(21,101)(22,100)(23,132)(24,131)(25,130)(26,129)(27,128)(28,127)(29,126)(30,125)(31,124)(32,123)(33,122)(34,121)(35,120)(36,119)(37,118)(38,117)(39,116)(40,115)(41,114)(42,113)(43,112)(44,111)(45,165)(46,164)(47,163)(48,162)(49,161)(50,160)(51,159)(52,158)(53,157)(54,156)(55,155)(56,176)(57,175)(58,174)(59,173)(60,172)(61,171)(62,170)(63,169)(64,168)(65,167)(66,166)(67,143)(68,142)(69,141)(70,140)(71,139)(72,138)(73,137)(74,136)(75,135)(76,134)(77,133)(78,154)(79,153)(80,152)(81,151)(82,150)(83,149)(84,148)(85,147)(86,146)(87,145)(88,144)>;

G:=Group( (1,89)(2,90)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,97)(10,98)(11,99)(12,100)(13,101)(14,102)(15,103)(16,104)(17,105)(18,106)(19,107)(20,108)(21,109)(22,110)(23,111)(24,112)(25,113)(26,114)(27,115)(28,116)(29,117)(30,118)(31,119)(32,120)(33,121)(34,122)(35,123)(36,124)(37,125)(38,126)(39,127)(40,128)(41,129)(42,130)(43,131)(44,132)(45,133)(46,134)(47,135)(48,136)(49,137)(50,138)(51,139)(52,140)(53,141)(54,142)(55,143)(56,144)(57,145)(58,146)(59,147)(60,148)(61,149)(62,150)(63,151)(64,152)(65,153)(66,154)(67,155)(68,156)(69,157)(70,158)(71,159)(72,160)(73,161)(74,162)(75,163)(76,164)(77,165)(78,166)(79,167)(80,168)(81,169)(82,170)(83,171)(84,172)(85,173)(86,174)(87,175)(88,176), (1,166,34,144,12,155,23,133)(2,167,35,145,13,156,24,134)(3,168,36,146,14,157,25,135)(4,169,37,147,15,158,26,136)(5,170,38,148,16,159,27,137)(6,171,39,149,17,160,28,138)(7,172,40,150,18,161,29,139)(8,173,41,151,19,162,30,140)(9,174,42,152,20,163,31,141)(10,175,43,153,21,164,32,142)(11,176,44,154,22,165,33,143)(45,89,78,122,56,100,67,111)(46,90,79,123,57,101,68,112)(47,91,80,124,58,102,69,113)(48,92,81,125,59,103,70,114)(49,93,82,126,60,104,71,115)(50,94,83,127,61,105,72,116)(51,95,84,128,62,106,73,117)(52,96,85,129,63,107,74,118)(53,97,86,130,64,108,75,119)(54,98,87,131,65,109,76,120)(55,99,88,132,66,110,77,121), (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), (1,99)(2,98)(3,97)(4,96)(5,95)(6,94)(7,93)(8,92)(9,91)(10,90)(11,89)(12,110)(13,109)(14,108)(15,107)(16,106)(17,105)(18,104)(19,103)(20,102)(21,101)(22,100)(23,132)(24,131)(25,130)(26,129)(27,128)(28,127)(29,126)(30,125)(31,124)(32,123)(33,122)(34,121)(35,120)(36,119)(37,118)(38,117)(39,116)(40,115)(41,114)(42,113)(43,112)(44,111)(45,165)(46,164)(47,163)(48,162)(49,161)(50,160)(51,159)(52,158)(53,157)(54,156)(55,155)(56,176)(57,175)(58,174)(59,173)(60,172)(61,171)(62,170)(63,169)(64,168)(65,167)(66,166)(67,143)(68,142)(69,141)(70,140)(71,139)(72,138)(73,137)(74,136)(75,135)(76,134)(77,133)(78,154)(79,153)(80,152)(81,151)(82,150)(83,149)(84,148)(85,147)(86,146)(87,145)(88,144) );

G=PermutationGroup([[(1,89),(2,90),(3,91),(4,92),(5,93),(6,94),(7,95),(8,96),(9,97),(10,98),(11,99),(12,100),(13,101),(14,102),(15,103),(16,104),(17,105),(18,106),(19,107),(20,108),(21,109),(22,110),(23,111),(24,112),(25,113),(26,114),(27,115),(28,116),(29,117),(30,118),(31,119),(32,120),(33,121),(34,122),(35,123),(36,124),(37,125),(38,126),(39,127),(40,128),(41,129),(42,130),(43,131),(44,132),(45,133),(46,134),(47,135),(48,136),(49,137),(50,138),(51,139),(52,140),(53,141),(54,142),(55,143),(56,144),(57,145),(58,146),(59,147),(60,148),(61,149),(62,150),(63,151),(64,152),(65,153),(66,154),(67,155),(68,156),(69,157),(70,158),(71,159),(72,160),(73,161),(74,162),(75,163),(76,164),(77,165),(78,166),(79,167),(80,168),(81,169),(82,170),(83,171),(84,172),(85,173),(86,174),(87,175),(88,176)], [(1,166,34,144,12,155,23,133),(2,167,35,145,13,156,24,134),(3,168,36,146,14,157,25,135),(4,169,37,147,15,158,26,136),(5,170,38,148,16,159,27,137),(6,171,39,149,17,160,28,138),(7,172,40,150,18,161,29,139),(8,173,41,151,19,162,30,140),(9,174,42,152,20,163,31,141),(10,175,43,153,21,164,32,142),(11,176,44,154,22,165,33,143),(45,89,78,122,56,100,67,111),(46,90,79,123,57,101,68,112),(47,91,80,124,58,102,69,113),(48,92,81,125,59,103,70,114),(49,93,82,126,60,104,71,115),(50,94,83,127,61,105,72,116),(51,95,84,128,62,106,73,117),(52,96,85,129,63,107,74,118),(53,97,86,130,64,108,75,119),(54,98,87,131,65,109,76,120),(55,99,88,132,66,110,77,121)], [(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)], [(1,99),(2,98),(3,97),(4,96),(5,95),(6,94),(7,93),(8,92),(9,91),(10,90),(11,89),(12,110),(13,109),(14,108),(15,107),(16,106),(17,105),(18,104),(19,103),(20,102),(21,101),(22,100),(23,132),(24,131),(25,130),(26,129),(27,128),(28,127),(29,126),(30,125),(31,124),(32,123),(33,122),(34,121),(35,120),(36,119),(37,118),(38,117),(39,116),(40,115),(41,114),(42,113),(43,112),(44,111),(45,165),(46,164),(47,163),(48,162),(49,161),(50,160),(51,159),(52,158),(53,157),(54,156),(55,155),(56,176),(57,175),(58,174),(59,173),(60,172),(61,171),(62,170),(63,169),(64,168),(65,167),(66,166),(67,143),(68,142),(69,141),(70,140),(71,139),(72,138),(73,137),(74,136),(75,135),(76,134),(77,133),(78,154),(79,153),(80,152),(81,151),(82,150),(83,149),(84,148),(85,147),(86,146),(87,145),(88,144)]])

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 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 D4 D4 SD16 D11 D22 D22 D44 D44 C8⋊D11 kernel C2×C8⋊D11 C8⋊D11 C2×C88 C2×Dic22 C2×D44 C44 C2×C22 C22 C2×C8 C8 C2×C4 C4 C22 C2 # reps 1 4 1 1 1 1 1 4 5 10 5 10 10 40

Matrix representation of C2×C8⋊D11 in GL3(𝔽89) generated by

 88 0 0 0 1 0 0 0 1
,
 88 0 0 0 9 23 0 66 40
,
 1 0 0 0 71 88 0 1 0
,
 1 0 0 0 71 88 0 56 18
G:=sub<GL(3,GF(89))| [88,0,0,0,1,0,0,0,1],[88,0,0,0,9,66,0,23,40],[1,0,0,0,71,1,0,88,0],[1,0,0,0,71,56,0,88,18] >;

C2×C8⋊D11 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes D_{11}
% in TeX

G:=Group("C2xC8:D11");
// GroupNames label

G:=SmallGroup(352,97);
// by ID

G=gap.SmallGroup(352,97);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-11,218,50,579,69,11525]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^11=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^3,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽