direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Dic22, C22⋊Q8, C4.11D22, C22.1C23, C22.8D22, C44.11C22, Dic11.1C22, C11⋊1(C2×Q8), (C2×C44).4C2, (C2×C4).4D11, (C2×C22).8C22, C2.3(C22×D11), (C2×Dic11).3C2, SmallGroup(176,27)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Dic22
G = < a,b,c | a2=b44=1, c2=b22, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C2×Dic22
►Regular action on 176 points
Generators in S176
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 73)(8 74)(9 75)(10 76)(11 77)(12 78)(13 79)(14 80)(15 81)(16 82)(17 83)(18 84)(19 85)(20 86)(21 87)(22 88)(23 45)(24 46)(25 47)(26 48)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 55)(34 56)(35 57)(36 58)(37 59)(38 60)(39 61)(40 62)(41 63)(42 64)(43 65)(44 66)(89 155)(90 156)(91 157)(92 158)(93 159)(94 160)(95 161)(96 162)(97 163)(98 164)(99 165)(100 166)(101 167)(102 168)(103 169)(104 170)(105 171)(106 172)(107 173)(108 174)(109 175)(110 176)(111 133)(112 134)(113 135)(114 136)(115 137)(116 138)(117 139)(118 140)(119 141)(120 142)(121 143)(122 144)(123 145)(124 146)(125 147)(126 148)(127 149)(128 150)(129 151)(130 152)(131 153)(132 154)
(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)
(1 122 23 100)(2 121 24 99)(3 120 25 98)(4 119 26 97)(5 118 27 96)(6 117 28 95)(7 116 29 94)(8 115 30 93)(9 114 31 92)(10 113 32 91)(11 112 33 90)(12 111 34 89)(13 110 35 132)(14 109 36 131)(15 108 37 130)(16 107 38 129)(17 106 39 128)(18 105 40 127)(19 104 41 126)(20 103 42 125)(21 102 43 124)(22 101 44 123)(45 166 67 144)(46 165 68 143)(47 164 69 142)(48 163 70 141)(49 162 71 140)(50 161 72 139)(51 160 73 138)(52 159 74 137)(53 158 75 136)(54 157 76 135)(55 156 77 134)(56 155 78 133)(57 154 79 176)(58 153 80 175)(59 152 81 174)(60 151 82 173)(61 150 83 172)(62 149 84 171)(63 148 85 170)(64 147 86 169)(65 146 87 168)(66 145 88 167)
G:=sub<Sym(176)| (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,73)(8,74)(9,75)(10,76)(11,77)(12,78)(13,79)(14,80)(15,81)(16,82)(17,83)(18,84)(19,85)(20,86)(21,87)(22,88)(23,45)(24,46)(25,47)(26,48)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,55)(34,56)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(41,63)(42,64)(43,65)(44,66)(89,155)(90,156)(91,157)(92,158)(93,159)(94,160)(95,161)(96,162)(97,163)(98,164)(99,165)(100,166)(101,167)(102,168)(103,169)(104,170)(105,171)(106,172)(107,173)(108,174)(109,175)(110,176)(111,133)(112,134)(113,135)(114,136)(115,137)(116,138)(117,139)(118,140)(119,141)(120,142)(121,143)(122,144)(123,145)(124,146)(125,147)(126,148)(127,149)(128,150)(129,151)(130,152)(131,153)(132,154), (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), (1,122,23,100)(2,121,24,99)(3,120,25,98)(4,119,26,97)(5,118,27,96)(6,117,28,95)(7,116,29,94)(8,115,30,93)(9,114,31,92)(10,113,32,91)(11,112,33,90)(12,111,34,89)(13,110,35,132)(14,109,36,131)(15,108,37,130)(16,107,38,129)(17,106,39,128)(18,105,40,127)(19,104,41,126)(20,103,42,125)(21,102,43,124)(22,101,44,123)(45,166,67,144)(46,165,68,143)(47,164,69,142)(48,163,70,141)(49,162,71,140)(50,161,72,139)(51,160,73,138)(52,159,74,137)(53,158,75,136)(54,157,76,135)(55,156,77,134)(56,155,78,133)(57,154,79,176)(58,153,80,175)(59,152,81,174)(60,151,82,173)(61,150,83,172)(62,149,84,171)(63,148,85,170)(64,147,86,169)(65,146,87,168)(66,145,88,167)>;
G:=Group( (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,73)(8,74)(9,75)(10,76)(11,77)(12,78)(13,79)(14,80)(15,81)(16,82)(17,83)(18,84)(19,85)(20,86)(21,87)(22,88)(23,45)(24,46)(25,47)(26,48)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,55)(34,56)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(41,63)(42,64)(43,65)(44,66)(89,155)(90,156)(91,157)(92,158)(93,159)(94,160)(95,161)(96,162)(97,163)(98,164)(99,165)(100,166)(101,167)(102,168)(103,169)(104,170)(105,171)(106,172)(107,173)(108,174)(109,175)(110,176)(111,133)(112,134)(113,135)(114,136)(115,137)(116,138)(117,139)(118,140)(119,141)(120,142)(121,143)(122,144)(123,145)(124,146)(125,147)(126,148)(127,149)(128,150)(129,151)(130,152)(131,153)(132,154), (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), (1,122,23,100)(2,121,24,99)(3,120,25,98)(4,119,26,97)(5,118,27,96)(6,117,28,95)(7,116,29,94)(8,115,30,93)(9,114,31,92)(10,113,32,91)(11,112,33,90)(12,111,34,89)(13,110,35,132)(14,109,36,131)(15,108,37,130)(16,107,38,129)(17,106,39,128)(18,105,40,127)(19,104,41,126)(20,103,42,125)(21,102,43,124)(22,101,44,123)(45,166,67,144)(46,165,68,143)(47,164,69,142)(48,163,70,141)(49,162,71,140)(50,161,72,139)(51,160,73,138)(52,159,74,137)(53,158,75,136)(54,157,76,135)(55,156,77,134)(56,155,78,133)(57,154,79,176)(58,153,80,175)(59,152,81,174)(60,151,82,173)(61,150,83,172)(62,149,84,171)(63,148,85,170)(64,147,86,169)(65,146,87,168)(66,145,88,167) );
G=PermutationGroup([[(1,67),(2,68),(3,69),(4,70),(5,71),(6,72),(7,73),(8,74),(9,75),(10,76),(11,77),(12,78),(13,79),(14,80),(15,81),(16,82),(17,83),(18,84),(19,85),(20,86),(21,87),(22,88),(23,45),(24,46),(25,47),(26,48),(27,49),(28,50),(29,51),(30,52),(31,53),(32,54),(33,55),(34,56),(35,57),(36,58),(37,59),(38,60),(39,61),(40,62),(41,63),(42,64),(43,65),(44,66),(89,155),(90,156),(91,157),(92,158),(93,159),(94,160),(95,161),(96,162),(97,163),(98,164),(99,165),(100,166),(101,167),(102,168),(103,169),(104,170),(105,171),(106,172),(107,173),(108,174),(109,175),(110,176),(111,133),(112,134),(113,135),(114,136),(115,137),(116,138),(117,139),(118,140),(119,141),(120,142),(121,143),(122,144),(123,145),(124,146),(125,147),(126,148),(127,149),(128,150),(129,151),(130,152),(131,153),(132,154)], [(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)], [(1,122,23,100),(2,121,24,99),(3,120,25,98),(4,119,26,97),(5,118,27,96),(6,117,28,95),(7,116,29,94),(8,115,30,93),(9,114,31,92),(10,113,32,91),(11,112,33,90),(12,111,34,89),(13,110,35,132),(14,109,36,131),(15,108,37,130),(16,107,38,129),(17,106,39,128),(18,105,40,127),(19,104,41,126),(20,103,42,125),(21,102,43,124),(22,101,44,123),(45,166,67,144),(46,165,68,143),(47,164,69,142),(48,163,70,141),(49,162,71,140),(50,161,72,139),(51,160,73,138),(52,159,74,137),(53,158,75,136),(54,157,76,135),(55,156,77,134),(56,155,78,133),(57,154,79,176),(58,153,80,175),(59,152,81,174),(60,151,82,173),(61,150,83,172),(62,149,84,171),(63,148,85,170),(64,147,86,169),(65,146,87,168),(66,145,88,167)]])
C2×Dic22 is a maximal subgroup of
C22.Q16 C44.44D4 C44.47D4 C44⋊2Q8 C4.D44 C22⋊Dic22 Dic11.D4 Dic22⋊C4 C44⋊Q8 D22⋊Q8 D22⋊2Q8 C8.D22 C44.48D4 C44.17D4 Dic11⋊Q8 D4.9D22 C2×Q8×D11 D4.10D22
C2×Dic22 is a maximal quotient of
C44⋊2Q8 C44.6Q8 C22⋊Dic22 C44⋊Q8 C44.3Q8 C44.48D4
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 11A | ··· | 11E | 22A | ··· | 22O | 44A | ··· | 44T |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 22 | 22 | 22 | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | + | - |
| image | C1 | C2 | C2 | C2 | Q8 | D11 | D22 | D22 | Dic22 |
| kernel | C2×Dic22 | Dic22 | C2×Dic11 | C2×C44 | C22 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 2 | 1 | 2 | 5 | 10 | 5 | 20 |
Matrix representation of C2×Dic22 ►in GL3(𝔽89) generated by
| 88 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 59 | 86 |
| 0 | 3 | 24 |
| 88 | 0 | 0 |
| 0 | 50 | 1 |
| 0 | 80 | 39 |
G:=sub<GL(3,GF(89))| [88,0,0,0,1,0,0,0,1],[1,0,0,0,59,3,0,86,24],[88,0,0,0,50,80,0,1,39] >;
C2×Dic22 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_{22} % in TeX
G:=Group("C2xDic22"); // GroupNames label
G:=SmallGroup(176,27);
// by ID
G=gap.SmallGroup(176,27);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-11,40,182,42,4004]);
// Polycyclic
G:=Group<a,b,c|a^2=b^44=1,c^2=b^22,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export