direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C2×C22, C24⋊2C22, C44⋊4C23, C22.16C24, C4⋊(C22×C22), (C2×C22)⋊2C23, (C23×C22)⋊2C2, (C22×C4)⋊5C22, C23⋊3(C2×C22), C22⋊(C22×C22), (C22×C44)⋊12C2, (C2×C44)⋊15C22, C2.1(C23×C22), (C22×C22)⋊6C22, (C2×C4)⋊4(C2×C22), SmallGroup(352,189)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C2×C22
G = < a,b,c,d | a2=b22=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 316 in 236 conjugacy classes, 156 normal (10 characteristic)
C1, C2, C2, C2, C4, C22, C22, C2×C4, D4, C23, C23, C23, C11, C22×C4, C2×D4, C24, C22, C22, C22, C22×D4, C44, C2×C22, C2×C22, C2×C44, D4×C11, C22×C22, C22×C22, C22×C22, C22×C44, D4×C22, C23×C22, D4×C2×C22
Quotients: C1, C2, C22, D4, C23, C11, C2×D4, C24, C22, C22×D4, C2×C22, D4×C11, C22×C22, D4×C22, C23×C22, D4×C2×C22
►On 176 points
Generators in S176
(1 134)(2 135)(3 136)(4 137)(5 138)(6 139)(7 140)(8 141)(9 142)(10 143)(11 144)(12 145)(13 146)(14 147)(15 148)(16 149)(17 150)(18 151)(19 152)(20 153)(21 154)(22 133)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 67)(42 68)(43 69)(44 70)(45 108)(46 109)(47 110)(48 89)(49 90)(50 91)(51 92)(52 93)(53 94)(54 95)(55 96)(56 97)(57 98)(58 99)(59 100)(60 101)(61 102)(62 103)(63 104)(64 105)(65 106)(66 107)(111 165)(112 166)(113 167)(114 168)(115 169)(116 170)(117 171)(118 172)(119 173)(120 174)(121 175)(122 176)(123 155)(124 156)(125 157)(126 158)(127 159)(128 160)(129 161)(130 162)(131 163)(132 164)
(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)
(1 167 28 90)(2 168 29 91)(3 169 30 92)(4 170 31 93)(5 171 32 94)(6 172 33 95)(7 173 34 96)(8 174 35 97)(9 175 36 98)(10 176 37 99)(11 155 38 100)(12 156 39 101)(13 157 40 102)(14 158 41 103)(15 159 42 104)(16 160 43 105)(17 161 44 106)(18 162 23 107)(19 163 24 108)(20 164 25 109)(21 165 26 110)(22 166 27 89)(45 152 131 72)(46 153 132 73)(47 154 111 74)(48 133 112 75)(49 134 113 76)(50 135 114 77)(51 136 115 78)(52 137 116 79)(53 138 117 80)(54 139 118 81)(55 140 119 82)(56 141 120 83)(57 142 121 84)(58 143 122 85)(59 144 123 86)(60 145 124 87)(61 146 125 88)(62 147 126 67)(63 148 127 68)(64 149 128 69)(65 150 129 70)(66 151 130 71)
(1 90)(2 91)(3 92)(4 93)(5 94)(6 95)(7 96)(8 97)(9 98)(10 99)(11 100)(12 101)(13 102)(14 103)(15 104)(16 105)(17 106)(18 107)(19 108)(20 109)(21 110)(22 89)(23 162)(24 163)(25 164)(26 165)(27 166)(28 167)(29 168)(30 169)(31 170)(32 171)(33 172)(34 173)(35 174)(36 175)(37 176)(38 155)(39 156)(40 157)(41 158)(42 159)(43 160)(44 161)(45 152)(46 153)(47 154)(48 133)(49 134)(50 135)(51 136)(52 137)(53 138)(54 139)(55 140)(56 141)(57 142)(58 143)(59 144)(60 145)(61 146)(62 147)(63 148)(64 149)(65 150)(66 151)(67 126)(68 127)(69 128)(70 129)(71 130)(72 131)(73 132)(74 111)(75 112)(76 113)(77 114)(78 115)(79 116)(80 117)(81 118)(82 119)(83 120)(84 121)(85 122)(86 123)(87 124)(88 125)
G:=sub<Sym(176)| (1,134)(2,135)(3,136)(4,137)(5,138)(6,139)(7,140)(8,141)(9,142)(10,143)(11,144)(12,145)(13,146)(14,147)(15,148)(16,149)(17,150)(18,151)(19,152)(20,153)(21,154)(22,133)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,67)(42,68)(43,69)(44,70)(45,108)(46,109)(47,110)(48,89)(49,90)(50,91)(51,92)(52,93)(53,94)(54,95)(55,96)(56,97)(57,98)(58,99)(59,100)(60,101)(61,102)(62,103)(63,104)(64,105)(65,106)(66,107)(111,165)(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)(118,172)(119,173)(120,174)(121,175)(122,176)(123,155)(124,156)(125,157)(126,158)(127,159)(128,160)(129,161)(130,162)(131,163)(132,164), (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), (1,167,28,90)(2,168,29,91)(3,169,30,92)(4,170,31,93)(5,171,32,94)(6,172,33,95)(7,173,34,96)(8,174,35,97)(9,175,36,98)(10,176,37,99)(11,155,38,100)(12,156,39,101)(13,157,40,102)(14,158,41,103)(15,159,42,104)(16,160,43,105)(17,161,44,106)(18,162,23,107)(19,163,24,108)(20,164,25,109)(21,165,26,110)(22,166,27,89)(45,152,131,72)(46,153,132,73)(47,154,111,74)(48,133,112,75)(49,134,113,76)(50,135,114,77)(51,136,115,78)(52,137,116,79)(53,138,117,80)(54,139,118,81)(55,140,119,82)(56,141,120,83)(57,142,121,84)(58,143,122,85)(59,144,123,86)(60,145,124,87)(61,146,125,88)(62,147,126,67)(63,148,127,68)(64,149,128,69)(65,150,129,70)(66,151,130,71), (1,90)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,100)(12,101)(13,102)(14,103)(15,104)(16,105)(17,106)(18,107)(19,108)(20,109)(21,110)(22,89)(23,162)(24,163)(25,164)(26,165)(27,166)(28,167)(29,168)(30,169)(31,170)(32,171)(33,172)(34,173)(35,174)(36,175)(37,176)(38,155)(39,156)(40,157)(41,158)(42,159)(43,160)(44,161)(45,152)(46,153)(47,154)(48,133)(49,134)(50,135)(51,136)(52,137)(53,138)(54,139)(55,140)(56,141)(57,142)(58,143)(59,144)(60,145)(61,146)(62,147)(63,148)(64,149)(65,150)(66,151)(67,126)(68,127)(69,128)(70,129)(71,130)(72,131)(73,132)(74,111)(75,112)(76,113)(77,114)(78,115)(79,116)(80,117)(81,118)(82,119)(83,120)(84,121)(85,122)(86,123)(87,124)(88,125)>;
G:=Group( (1,134)(2,135)(3,136)(4,137)(5,138)(6,139)(7,140)(8,141)(9,142)(10,143)(11,144)(12,145)(13,146)(14,147)(15,148)(16,149)(17,150)(18,151)(19,152)(20,153)(21,154)(22,133)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,67)(42,68)(43,69)(44,70)(45,108)(46,109)(47,110)(48,89)(49,90)(50,91)(51,92)(52,93)(53,94)(54,95)(55,96)(56,97)(57,98)(58,99)(59,100)(60,101)(61,102)(62,103)(63,104)(64,105)(65,106)(66,107)(111,165)(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)(118,172)(119,173)(120,174)(121,175)(122,176)(123,155)(124,156)(125,157)(126,158)(127,159)(128,160)(129,161)(130,162)(131,163)(132,164), (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), (1,167,28,90)(2,168,29,91)(3,169,30,92)(4,170,31,93)(5,171,32,94)(6,172,33,95)(7,173,34,96)(8,174,35,97)(9,175,36,98)(10,176,37,99)(11,155,38,100)(12,156,39,101)(13,157,40,102)(14,158,41,103)(15,159,42,104)(16,160,43,105)(17,161,44,106)(18,162,23,107)(19,163,24,108)(20,164,25,109)(21,165,26,110)(22,166,27,89)(45,152,131,72)(46,153,132,73)(47,154,111,74)(48,133,112,75)(49,134,113,76)(50,135,114,77)(51,136,115,78)(52,137,116,79)(53,138,117,80)(54,139,118,81)(55,140,119,82)(56,141,120,83)(57,142,121,84)(58,143,122,85)(59,144,123,86)(60,145,124,87)(61,146,125,88)(62,147,126,67)(63,148,127,68)(64,149,128,69)(65,150,129,70)(66,151,130,71), (1,90)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,97)(9,98)(10,99)(11,100)(12,101)(13,102)(14,103)(15,104)(16,105)(17,106)(18,107)(19,108)(20,109)(21,110)(22,89)(23,162)(24,163)(25,164)(26,165)(27,166)(28,167)(29,168)(30,169)(31,170)(32,171)(33,172)(34,173)(35,174)(36,175)(37,176)(38,155)(39,156)(40,157)(41,158)(42,159)(43,160)(44,161)(45,152)(46,153)(47,154)(48,133)(49,134)(50,135)(51,136)(52,137)(53,138)(54,139)(55,140)(56,141)(57,142)(58,143)(59,144)(60,145)(61,146)(62,147)(63,148)(64,149)(65,150)(66,151)(67,126)(68,127)(69,128)(70,129)(71,130)(72,131)(73,132)(74,111)(75,112)(76,113)(77,114)(78,115)(79,116)(80,117)(81,118)(82,119)(83,120)(84,121)(85,122)(86,123)(87,124)(88,125) );
G=PermutationGroup([[(1,134),(2,135),(3,136),(4,137),(5,138),(6,139),(7,140),(8,141),(9,142),(10,143),(11,144),(12,145),(13,146),(14,147),(15,148),(16,149),(17,150),(18,151),(19,152),(20,153),(21,154),(22,133),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,77),(30,78),(31,79),(32,80),(33,81),(34,82),(35,83),(36,84),(37,85),(38,86),(39,87),(40,88),(41,67),(42,68),(43,69),(44,70),(45,108),(46,109),(47,110),(48,89),(49,90),(50,91),(51,92),(52,93),(53,94),(54,95),(55,96),(56,97),(57,98),(58,99),(59,100),(60,101),(61,102),(62,103),(63,104),(64,105),(65,106),(66,107),(111,165),(112,166),(113,167),(114,168),(115,169),(116,170),(117,171),(118,172),(119,173),(120,174),(121,175),(122,176),(123,155),(124,156),(125,157),(126,158),(127,159),(128,160),(129,161),(130,162),(131,163),(132,164)], [(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)], [(1,167,28,90),(2,168,29,91),(3,169,30,92),(4,170,31,93),(5,171,32,94),(6,172,33,95),(7,173,34,96),(8,174,35,97),(9,175,36,98),(10,176,37,99),(11,155,38,100),(12,156,39,101),(13,157,40,102),(14,158,41,103),(15,159,42,104),(16,160,43,105),(17,161,44,106),(18,162,23,107),(19,163,24,108),(20,164,25,109),(21,165,26,110),(22,166,27,89),(45,152,131,72),(46,153,132,73),(47,154,111,74),(48,133,112,75),(49,134,113,76),(50,135,114,77),(51,136,115,78),(52,137,116,79),(53,138,117,80),(54,139,118,81),(55,140,119,82),(56,141,120,83),(57,142,121,84),(58,143,122,85),(59,144,123,86),(60,145,124,87),(61,146,125,88),(62,147,126,67),(63,148,127,68),(64,149,128,69),(65,150,129,70),(66,151,130,71)], [(1,90),(2,91),(3,92),(4,93),(5,94),(6,95),(7,96),(8,97),(9,98),(10,99),(11,100),(12,101),(13,102),(14,103),(15,104),(16,105),(17,106),(18,107),(19,108),(20,109),(21,110),(22,89),(23,162),(24,163),(25,164),(26,165),(27,166),(28,167),(29,168),(30,169),(31,170),(32,171),(33,172),(34,173),(35,174),(36,175),(37,176),(38,155),(39,156),(40,157),(41,158),(42,159),(43,160),(44,161),(45,152),(46,153),(47,154),(48,133),(49,134),(50,135),(51,136),(52,137),(53,138),(54,139),(55,140),(56,141),(57,142),(58,143),(59,144),(60,145),(61,146),(62,147),(63,148),(64,149),(65,150),(66,151),(67,126),(68,127),(69,128),(70,129),(71,130),(72,131),(73,132),(74,111),(75,112),(76,113),(77,114),(78,115),(79,116),(80,117),(81,118),(82,119),(83,120),(84,121),(85,122),(86,123),(87,124),(88,125)]])
220 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 4A | 4B | 4C | 4D | 11A | ··· | 11J | 22A | ··· | 22BR | 22BS | ··· | 22ET | 44A | ··· | 44AN |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
220 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C11 | C22 | C22 | C22 | D4 | D4×C11 |
| kernel | D4×C2×C22 | C22×C44 | D4×C22 | C23×C22 | C22×D4 | C22×C4 | C2×D4 | C24 | C2×C22 | C22 |
| # reps | 1 | 1 | 12 | 2 | 10 | 10 | 120 | 20 | 4 | 40 |
Matrix representation of D4×C2×C22 ►in GL4(𝔽89) generated by
| 88 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 39 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 87 | 0 |
| 0 | 0 | 0 | 87 |
| 1 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 1 | 87 |
| 0 | 0 | 1 | 88 |
| 88 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 87 |
| 0 | 0 | 0 | 88 |
G:=sub<GL(4,GF(89))| [88,0,0,0,0,88,0,0,0,0,1,0,0,0,0,1],[39,0,0,0,0,88,0,0,0,0,87,0,0,0,0,87],[1,0,0,0,0,88,0,0,0,0,1,1,0,0,87,88],[88,0,0,0,0,1,0,0,0,0,1,0,0,0,87,88] >;
D4×C2×C22 in GAP, Magma, Sage, TeX
D_4\times C_2\times C_{22} % in TeX
G:=Group("D4xC2xC22"); // GroupNames label
G:=SmallGroup(352,189);
// by ID
G=gap.SmallGroup(352,189);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-11,-2,2137]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^22=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations