direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C8⋊D11, C8⋊8D22, C4.6D44, C88⋊9C22, C22⋊1SD16, C44.29D4, C44.28C23, D44.6C22, C22.12D44, Dic22⋊3C22, (C2×C88)⋊7C2, (C2×C8)⋊5D11, C22.9(C2×D4), C11⋊1(C2×SD16), (C2×D44).4C2, C2.11(C2×D44), (C2×C22).16D4, (C2×C4).79D22, (C2×Dic22)⋊5C2, (C2×C44).88C22, C4.26(C22×D11), SmallGroup(352,97)
Series: Derived ►Chief ►Lower central ►Upper central
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
►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