direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C11⋊C8, C22⋊C8, C44.3C4, C4.14D22, C4.3Dic11, C44.14C22, C22.2Dic11, C11⋊2(C2×C8), (C2×C44).6C2, C22.6(C2×C4), (C2×C22).2C4, (C2×C4).5D11, C2.1(C2×Dic11), SmallGroup(176,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C11 — C2×C11⋊C8 |
Generators and relations for C2×C11⋊C8
G = < a,b,c | a2=b11=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×C11⋊C8
►Regular action on 176 points
Generators in S176
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 71)(28 72)(29 73)(30 74)(31 75)(32 76)(33 77)(34 78)(35 79)(36 80)(37 81)(38 82)(39 83)(40 84)(41 85)(42 86)(43 87)(44 88)(89 133)(90 134)(91 135)(92 136)(93 137)(94 138)(95 139)(96 140)(97 141)(98 142)(99 143)(100 144)(101 145)(102 146)(103 147)(104 148)(105 149)(106 150)(107 151)(108 152)(109 153)(110 154)(111 155)(112 156)(113 157)(114 158)(115 159)(116 160)(117 161)(118 162)(119 163)(120 164)(121 165)(122 166)(123 167)(124 168)(125 169)(126 170)(127 171)(128 172)(129 173)(130 174)(131 175)(132 176)
(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)
(1 175 34 153 12 164 23 142)(2 174 35 152 13 163 24 141)(3 173 36 151 14 162 25 140)(4 172 37 150 15 161 26 139)(5 171 38 149 16 160 27 138)(6 170 39 148 17 159 28 137)(7 169 40 147 18 158 29 136)(8 168 41 146 19 157 30 135)(9 167 42 145 20 156 31 134)(10 166 43 144 21 155 32 133)(11 176 44 154 22 165 33 143)(45 131 78 109 56 120 67 98)(46 130 79 108 57 119 68 97)(47 129 80 107 58 118 69 96)(48 128 81 106 59 117 70 95)(49 127 82 105 60 116 71 94)(50 126 83 104 61 115 72 93)(51 125 84 103 62 114 73 92)(52 124 85 102 63 113 74 91)(53 123 86 101 64 112 75 90)(54 122 87 100 65 111 76 89)(55 132 88 110 66 121 77 99)
G:=sub<Sym(176)| (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,61)(18,62)(19,63)(20,64)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(37,81)(38,82)(39,83)(40,84)(41,85)(42,86)(43,87)(44,88)(89,133)(90,134)(91,135)(92,136)(93,137)(94,138)(95,139)(96,140)(97,141)(98,142)(99,143)(100,144)(101,145)(102,146)(103,147)(104,148)(105,149)(106,150)(107,151)(108,152)(109,153)(110,154)(111,155)(112,156)(113,157)(114,158)(115,159)(116,160)(117,161)(118,162)(119,163)(120,164)(121,165)(122,166)(123,167)(124,168)(125,169)(126,170)(127,171)(128,172)(129,173)(130,174)(131,175)(132,176), (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), (1,175,34,153,12,164,23,142)(2,174,35,152,13,163,24,141)(3,173,36,151,14,162,25,140)(4,172,37,150,15,161,26,139)(5,171,38,149,16,160,27,138)(6,170,39,148,17,159,28,137)(7,169,40,147,18,158,29,136)(8,168,41,146,19,157,30,135)(9,167,42,145,20,156,31,134)(10,166,43,144,21,155,32,133)(11,176,44,154,22,165,33,143)(45,131,78,109,56,120,67,98)(46,130,79,108,57,119,68,97)(47,129,80,107,58,118,69,96)(48,128,81,106,59,117,70,95)(49,127,82,105,60,116,71,94)(50,126,83,104,61,115,72,93)(51,125,84,103,62,114,73,92)(52,124,85,102,63,113,74,91)(53,123,86,101,64,112,75,90)(54,122,87,100,65,111,76,89)(55,132,88,110,66,121,77,99)>;
G:=Group( (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,57)(14,58)(15,59)(16,60)(17,61)(18,62)(19,63)(20,64)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(37,81)(38,82)(39,83)(40,84)(41,85)(42,86)(43,87)(44,88)(89,133)(90,134)(91,135)(92,136)(93,137)(94,138)(95,139)(96,140)(97,141)(98,142)(99,143)(100,144)(101,145)(102,146)(103,147)(104,148)(105,149)(106,150)(107,151)(108,152)(109,153)(110,154)(111,155)(112,156)(113,157)(114,158)(115,159)(116,160)(117,161)(118,162)(119,163)(120,164)(121,165)(122,166)(123,167)(124,168)(125,169)(126,170)(127,171)(128,172)(129,173)(130,174)(131,175)(132,176), (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), (1,175,34,153,12,164,23,142)(2,174,35,152,13,163,24,141)(3,173,36,151,14,162,25,140)(4,172,37,150,15,161,26,139)(5,171,38,149,16,160,27,138)(6,170,39,148,17,159,28,137)(7,169,40,147,18,158,29,136)(8,168,41,146,19,157,30,135)(9,167,42,145,20,156,31,134)(10,166,43,144,21,155,32,133)(11,176,44,154,22,165,33,143)(45,131,78,109,56,120,67,98)(46,130,79,108,57,119,68,97)(47,129,80,107,58,118,69,96)(48,128,81,106,59,117,70,95)(49,127,82,105,60,116,71,94)(50,126,83,104,61,115,72,93)(51,125,84,103,62,114,73,92)(52,124,85,102,63,113,74,91)(53,123,86,101,64,112,75,90)(54,122,87,100,65,111,76,89)(55,132,88,110,66,121,77,99) );
G=PermutationGroup([[(1,45),(2,46),(3,47),(4,48),(5,49),(6,50),(7,51),(8,52),(9,53),(10,54),(11,55),(12,56),(13,57),(14,58),(15,59),(16,60),(17,61),(18,62),(19,63),(20,64),(21,65),(22,66),(23,67),(24,68),(25,69),(26,70),(27,71),(28,72),(29,73),(30,74),(31,75),(32,76),(33,77),(34,78),(35,79),(36,80),(37,81),(38,82),(39,83),(40,84),(41,85),(42,86),(43,87),(44,88),(89,133),(90,134),(91,135),(92,136),(93,137),(94,138),(95,139),(96,140),(97,141),(98,142),(99,143),(100,144),(101,145),(102,146),(103,147),(104,148),(105,149),(106,150),(107,151),(108,152),(109,153),(110,154),(111,155),(112,156),(113,157),(114,158),(115,159),(116,160),(117,161),(118,162),(119,163),(120,164),(121,165),(122,166),(123,167),(124,168),(125,169),(126,170),(127,171),(128,172),(129,173),(130,174),(131,175),(132,176)], [(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)], [(1,175,34,153,12,164,23,142),(2,174,35,152,13,163,24,141),(3,173,36,151,14,162,25,140),(4,172,37,150,15,161,26,139),(5,171,38,149,16,160,27,138),(6,170,39,148,17,159,28,137),(7,169,40,147,18,158,29,136),(8,168,41,146,19,157,30,135),(9,167,42,145,20,156,31,134),(10,166,43,144,21,155,32,133),(11,176,44,154,22,165,33,143),(45,131,78,109,56,120,67,98),(46,130,79,108,57,119,68,97),(47,129,80,107,58,118,69,96),(48,128,81,106,59,117,70,95),(49,127,82,105,60,116,71,94),(50,126,83,104,61,115,72,93),(51,125,84,103,62,114,73,92),(52,124,85,102,63,113,74,91),(53,123,86,101,64,112,75,90),(54,122,87,100,65,111,76,89),(55,132,88,110,66,121,77,99)]])
C2×C11⋊C8 is a maximal subgroup of
C42.D11 C44⋊C8 C44.Q8 C4.Dic22 C22.D8 C22.Q16 C8×Dic11 Dic11⋊C8 C88⋊C4 D22⋊C8 C44.53D4 C44.55D4 D4⋊Dic11 Q8⋊Dic11 C2×C8×D11 D44.C4 Q8.Dic11 D4.8D22
C2×C11⋊C8 is a maximal quotient of
C44⋊C8 C44.C8 C44.55D4
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 8A | ··· | 8H | 11A | ··· | 11E | 22A | ··· | 22O | 44A | ··· | 44T |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 11 | ··· | 11 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D11 | Dic11 | D22 | Dic11 | C11⋊C8 |
| kernel | C2×C11⋊C8 | C11⋊C8 | C2×C44 | C44 | C2×C22 | C22 | C2×C4 | C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 5 | 5 | 5 | 5 | 20 |
Matrix representation of C2×C11⋊C8 ►in GL3(𝔽89) generated by
| 1 | 0 | 0 |
| 0 | 88 | 0 |
| 0 | 0 | 88 |
| 1 | 0 | 0 |
| 0 | 55 | 88 |
| 0 | 1 | 0 |
| 77 | 0 | 0 |
| 0 | 31 | 47 |
| 0 | 61 | 58 |
G:=sub<GL(3,GF(89))| [1,0,0,0,88,0,0,0,88],[1,0,0,0,55,1,0,88,0],[77,0,0,0,31,61,0,47,58] >;
C2×C11⋊C8 in GAP, Magma, Sage, TeX
C_2\times C_{11}\rtimes C_8 % in TeX
G:=Group("C2xC11:C8"); // GroupNames label
G:=SmallGroup(176,8);
// by ID
G=gap.SmallGroup(176,8);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-11,20,42,4004]);
// Polycyclic
G:=Group<a,b,c|a^2=b^11=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export