direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D22⋊C4, C23.31D22, C22.16D44, (C2×C4)⋊8D22, D22⋊6(C2×C4), C2.3(C2×D44), (C22×C44)⋊1C2, (C2×C22).36D4, C22.40(C2×D4), (C22×C4)⋊1D11, C22⋊1(C22⋊C4), (C2×C44)⋊10C22, (C22×D11)⋊3C4, (C2×C22).45C23, C22.18(C22×C4), (C23×D11).2C2, C22.17(C4×D11), (C22×Dic11)⋊3C2, (C2×Dic11)⋊6C22, C22.20(C11⋊D4), (C22×C22).37C22, C22.23(C22×D11), (C22×D11).23C22, C11⋊2(C2×C22⋊C4), C2.19(C2×C4×D11), C2.2(C2×C11⋊D4), (C2×C22).18(C2×C4), SmallGroup(352,122)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D22⋊C4
G = < a,b,c,d | a2=b22=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b11c >
Subgroups: 858 in 132 conjugacy classes, 57 normal (17 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C11, C22⋊C4, C22×C4, C22×C4, C24, D11, C22, C22, C2×C22⋊C4, Dic11, C44, D22, D22, C2×C22, C2×C22, C2×Dic11, C2×Dic11, C2×C44, C2×C44, C22×D11, C22×D11, C22×C22, D22⋊C4, C22×Dic11, C22×C44, C23×D11, C2×D22⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, D11, C2×C22⋊C4, D22, C4×D11, D44, C11⋊D4, C22×D11, D22⋊C4, C2×C4×D11, C2×D44, C2×C11⋊D4, C2×D22⋊C4
►On 176 points
Generators in S176
(1 115)(2 116)(3 117)(4 118)(5 119)(6 120)(7 121)(8 122)(9 123)(10 124)(11 125)(12 126)(13 127)(14 128)(15 129)(16 130)(17 131)(18 132)(19 111)(20 112)(21 113)(22 114)(23 101)(24 102)(25 103)(26 104)(27 105)(28 106)(29 107)(30 108)(31 109)(32 110)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 97)(42 98)(43 99)(44 100)(45 172)(46 173)(47 174)(48 175)(49 176)(50 155)(51 156)(52 157)(53 158)(54 159)(55 160)(56 161)(57 162)(58 163)(59 164)(60 165)(61 166)(62 167)(63 168)(64 169)(65 170)(66 171)(67 144)(68 145)(69 146)(70 147)(71 148)(72 149)(73 150)(74 151)(75 152)(76 153)(77 154)(78 133)(79 134)(80 135)(81 136)(82 137)(83 138)(84 139)(85 140)(86 141)(87 142)(88 143)
(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)
(1 125)(2 124)(3 123)(4 122)(5 121)(6 120)(7 119)(8 118)(9 117)(10 116)(11 115)(12 114)(13 113)(14 112)(15 111)(16 132)(17 131)(18 130)(19 129)(20 128)(21 127)(22 126)(23 93)(24 92)(25 91)(26 90)(27 89)(28 110)(29 109)(30 108)(31 107)(32 106)(33 105)(34 104)(35 103)(36 102)(37 101)(38 100)(39 99)(40 98)(41 97)(42 96)(43 95)(44 94)(45 159)(46 158)(47 157)(48 156)(49 155)(50 176)(51 175)(52 174)(53 173)(54 172)(55 171)(56 170)(57 169)(58 168)(59 167)(60 166)(61 165)(62 164)(63 163)(64 162)(65 161)(66 160)(67 145)(68 144)(69 143)(70 142)(71 141)(72 140)(73 139)(74 138)(75 137)(76 136)(77 135)(78 134)(79 133)(80 154)(81 153)(82 152)(83 151)(84 150)(85 149)(86 148)(87 147)(88 146)
(1 79 36 61)(2 80 37 62)(3 81 38 63)(4 82 39 64)(5 83 40 65)(6 84 41 66)(7 85 42 45)(8 86 43 46)(9 87 44 47)(10 88 23 48)(11 67 24 49)(12 68 25 50)(13 69 26 51)(14 70 27 52)(15 71 28 53)(16 72 29 54)(17 73 30 55)(18 74 31 56)(19 75 32 57)(20 76 33 58)(21 77 34 59)(22 78 35 60)(89 163 112 153)(90 164 113 154)(91 165 114 133)(92 166 115 134)(93 167 116 135)(94 168 117 136)(95 169 118 137)(96 170 119 138)(97 171 120 139)(98 172 121 140)(99 173 122 141)(100 174 123 142)(101 175 124 143)(102 176 125 144)(103 155 126 145)(104 156 127 146)(105 157 128 147)(106 158 129 148)(107 159 130 149)(108 160 131 150)(109 161 132 151)(110 162 111 152)
G:=sub<Sym(176)| (1,115)(2,116)(3,117)(4,118)(5,119)(6,120)(7,121)(8,122)(9,123)(10,124)(11,125)(12,126)(13,127)(14,128)(15,129)(16,130)(17,131)(18,132)(19,111)(20,112)(21,113)(22,114)(23,101)(24,102)(25,103)(26,104)(27,105)(28,106)(29,107)(30,108)(31,109)(32,110)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(43,99)(44,100)(45,172)(46,173)(47,174)(48,175)(49,176)(50,155)(51,156)(52,157)(53,158)(54,159)(55,160)(56,161)(57,162)(58,163)(59,164)(60,165)(61,166)(62,167)(63,168)(64,169)(65,170)(66,171)(67,144)(68,145)(69,146)(70,147)(71,148)(72,149)(73,150)(74,151)(75,152)(76,153)(77,154)(78,133)(79,134)(80,135)(81,136)(82,137)(83,138)(84,139)(85,140)(86,141)(87,142)(88,143), (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), (1,125)(2,124)(3,123)(4,122)(5,121)(6,120)(7,119)(8,118)(9,117)(10,116)(11,115)(12,114)(13,113)(14,112)(15,111)(16,132)(17,131)(18,130)(19,129)(20,128)(21,127)(22,126)(23,93)(24,92)(25,91)(26,90)(27,89)(28,110)(29,109)(30,108)(31,107)(32,106)(33,105)(34,104)(35,103)(36,102)(37,101)(38,100)(39,99)(40,98)(41,97)(42,96)(43,95)(44,94)(45,159)(46,158)(47,157)(48,156)(49,155)(50,176)(51,175)(52,174)(53,173)(54,172)(55,171)(56,170)(57,169)(58,168)(59,167)(60,166)(61,165)(62,164)(63,163)(64,162)(65,161)(66,160)(67,145)(68,144)(69,143)(70,142)(71,141)(72,140)(73,139)(74,138)(75,137)(76,136)(77,135)(78,134)(79,133)(80,154)(81,153)(82,152)(83,151)(84,150)(85,149)(86,148)(87,147)(88,146), (1,79,36,61)(2,80,37,62)(3,81,38,63)(4,82,39,64)(5,83,40,65)(6,84,41,66)(7,85,42,45)(8,86,43,46)(9,87,44,47)(10,88,23,48)(11,67,24,49)(12,68,25,50)(13,69,26,51)(14,70,27,52)(15,71,28,53)(16,72,29,54)(17,73,30,55)(18,74,31,56)(19,75,32,57)(20,76,33,58)(21,77,34,59)(22,78,35,60)(89,163,112,153)(90,164,113,154)(91,165,114,133)(92,166,115,134)(93,167,116,135)(94,168,117,136)(95,169,118,137)(96,170,119,138)(97,171,120,139)(98,172,121,140)(99,173,122,141)(100,174,123,142)(101,175,124,143)(102,176,125,144)(103,155,126,145)(104,156,127,146)(105,157,128,147)(106,158,129,148)(107,159,130,149)(108,160,131,150)(109,161,132,151)(110,162,111,152)>;
G:=Group( (1,115)(2,116)(3,117)(4,118)(5,119)(6,120)(7,121)(8,122)(9,123)(10,124)(11,125)(12,126)(13,127)(14,128)(15,129)(16,130)(17,131)(18,132)(19,111)(20,112)(21,113)(22,114)(23,101)(24,102)(25,103)(26,104)(27,105)(28,106)(29,107)(30,108)(31,109)(32,110)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(43,99)(44,100)(45,172)(46,173)(47,174)(48,175)(49,176)(50,155)(51,156)(52,157)(53,158)(54,159)(55,160)(56,161)(57,162)(58,163)(59,164)(60,165)(61,166)(62,167)(63,168)(64,169)(65,170)(66,171)(67,144)(68,145)(69,146)(70,147)(71,148)(72,149)(73,150)(74,151)(75,152)(76,153)(77,154)(78,133)(79,134)(80,135)(81,136)(82,137)(83,138)(84,139)(85,140)(86,141)(87,142)(88,143), (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), (1,125)(2,124)(3,123)(4,122)(5,121)(6,120)(7,119)(8,118)(9,117)(10,116)(11,115)(12,114)(13,113)(14,112)(15,111)(16,132)(17,131)(18,130)(19,129)(20,128)(21,127)(22,126)(23,93)(24,92)(25,91)(26,90)(27,89)(28,110)(29,109)(30,108)(31,107)(32,106)(33,105)(34,104)(35,103)(36,102)(37,101)(38,100)(39,99)(40,98)(41,97)(42,96)(43,95)(44,94)(45,159)(46,158)(47,157)(48,156)(49,155)(50,176)(51,175)(52,174)(53,173)(54,172)(55,171)(56,170)(57,169)(58,168)(59,167)(60,166)(61,165)(62,164)(63,163)(64,162)(65,161)(66,160)(67,145)(68,144)(69,143)(70,142)(71,141)(72,140)(73,139)(74,138)(75,137)(76,136)(77,135)(78,134)(79,133)(80,154)(81,153)(82,152)(83,151)(84,150)(85,149)(86,148)(87,147)(88,146), (1,79,36,61)(2,80,37,62)(3,81,38,63)(4,82,39,64)(5,83,40,65)(6,84,41,66)(7,85,42,45)(8,86,43,46)(9,87,44,47)(10,88,23,48)(11,67,24,49)(12,68,25,50)(13,69,26,51)(14,70,27,52)(15,71,28,53)(16,72,29,54)(17,73,30,55)(18,74,31,56)(19,75,32,57)(20,76,33,58)(21,77,34,59)(22,78,35,60)(89,163,112,153)(90,164,113,154)(91,165,114,133)(92,166,115,134)(93,167,116,135)(94,168,117,136)(95,169,118,137)(96,170,119,138)(97,171,120,139)(98,172,121,140)(99,173,122,141)(100,174,123,142)(101,175,124,143)(102,176,125,144)(103,155,126,145)(104,156,127,146)(105,157,128,147)(106,158,129,148)(107,159,130,149)(108,160,131,150)(109,161,132,151)(110,162,111,152) );
G=PermutationGroup([[(1,115),(2,116),(3,117),(4,118),(5,119),(6,120),(7,121),(8,122),(9,123),(10,124),(11,125),(12,126),(13,127),(14,128),(15,129),(16,130),(17,131),(18,132),(19,111),(20,112),(21,113),(22,114),(23,101),(24,102),(25,103),(26,104),(27,105),(28,106),(29,107),(30,108),(31,109),(32,110),(33,89),(34,90),(35,91),(36,92),(37,93),(38,94),(39,95),(40,96),(41,97),(42,98),(43,99),(44,100),(45,172),(46,173),(47,174),(48,175),(49,176),(50,155),(51,156),(52,157),(53,158),(54,159),(55,160),(56,161),(57,162),(58,163),(59,164),(60,165),(61,166),(62,167),(63,168),(64,169),(65,170),(66,171),(67,144),(68,145),(69,146),(70,147),(71,148),(72,149),(73,150),(74,151),(75,152),(76,153),(77,154),(78,133),(79,134),(80,135),(81,136),(82,137),(83,138),(84,139),(85,140),(86,141),(87,142),(88,143)], [(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)], [(1,125),(2,124),(3,123),(4,122),(5,121),(6,120),(7,119),(8,118),(9,117),(10,116),(11,115),(12,114),(13,113),(14,112),(15,111),(16,132),(17,131),(18,130),(19,129),(20,128),(21,127),(22,126),(23,93),(24,92),(25,91),(26,90),(27,89),(28,110),(29,109),(30,108),(31,107),(32,106),(33,105),(34,104),(35,103),(36,102),(37,101),(38,100),(39,99),(40,98),(41,97),(42,96),(43,95),(44,94),(45,159),(46,158),(47,157),(48,156),(49,155),(50,176),(51,175),(52,174),(53,173),(54,172),(55,171),(56,170),(57,169),(58,168),(59,167),(60,166),(61,165),(62,164),(63,163),(64,162),(65,161),(66,160),(67,145),(68,144),(69,143),(70,142),(71,141),(72,140),(73,139),(74,138),(75,137),(76,136),(77,135),(78,134),(79,133),(80,154),(81,153),(82,152),(83,151),(84,150),(85,149),(86,148),(87,147),(88,146)], [(1,79,36,61),(2,80,37,62),(3,81,38,63),(4,82,39,64),(5,83,40,65),(6,84,41,66),(7,85,42,45),(8,86,43,46),(9,87,44,47),(10,88,23,48),(11,67,24,49),(12,68,25,50),(13,69,26,51),(14,70,27,52),(15,71,28,53),(16,72,29,54),(17,73,30,55),(18,74,31,56),(19,75,32,57),(20,76,33,58),(21,77,34,59),(22,78,35,60),(89,163,112,153),(90,164,113,154),(91,165,114,133),(92,166,115,134),(93,167,116,135),(94,168,117,136),(95,169,118,137),(96,170,119,138),(97,171,120,139),(98,172,121,140),(99,173,122,141),(100,174,123,142),(101,175,124,143),(102,176,125,144),(103,155,126,145),(104,156,127,146),(105,157,128,147),(106,158,129,148),(107,159,130,149),(108,160,131,150),(109,161,132,151),(110,162,111,152)]])
100 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 11A | ··· | 11E | 22A | ··· | 22AI | 44A | ··· | 44AN |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | ··· | 1 | 22 | 22 | 22 | 22 | 2 | 2 | 2 | 2 | 22 | 22 | 22 | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D11 | D22 | D22 | C4×D11 | D44 | C11⋊D4 |
| kernel | C2×D22⋊C4 | D22⋊C4 | C22×Dic11 | C22×C44 | C23×D11 | C22×D11 | C2×C22 | C22×C4 | C2×C4 | C23 | C22 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 4 | 5 | 10 | 5 | 20 | 20 | 20 |
Matrix representation of C2×D22⋊C4 ►in GL4(𝔽89) generated by
| 88 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 88 | 0 |
| 0 | 0 | 0 | 88 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 38 | 51 |
| 0 | 0 | 38 | 44 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 44 | 72 |
| 0 | 0 | 51 | 45 |
| 55 | 0 | 0 | 0 |
| 0 | 88 | 0 | 0 |
| 0 | 0 | 61 | 31 |
| 0 | 0 | 58 | 28 |
G:=sub<GL(4,GF(89))| [88,0,0,0,0,88,0,0,0,0,88,0,0,0,0,88],[1,0,0,0,0,1,0,0,0,0,38,38,0,0,51,44],[1,0,0,0,0,1,0,0,0,0,44,51,0,0,72,45],[55,0,0,0,0,88,0,0,0,0,61,58,0,0,31,28] >;
C2×D22⋊C4 in GAP, Magma, Sage, TeX
C_2\times D_{22}\rtimes C_4 % in TeX
G:=Group("C2xD22:C4"); // GroupNames label
G:=SmallGroup(352,122);
// by ID
G=gap.SmallGroup(352,122);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,362,50,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^22=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,b*d=d*b,d*c*d^-1=b^11*c>;
// generators/relations