direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D44⋊5C2, C22.4C24, D44⋊12C22, C44.43C23, D22.1C23, C23.26D22, Dic22⋊11C22, Dic11.2C23, (C2×C4)⋊10D22, (C2×D44)⋊14C2, C22⋊1(C4○D4), (C22×C44)⋊8C2, (C22×C4)⋊6D11, (C2×C44)⋊13C22, (C4×D11)⋊6C22, C11⋊D4⋊6C22, C2.5(C23×D11), (C2×Dic22)⋊15C2, (C2×C22).65C23, C4.43(C22×D11), C22.5(C22×D11), (C22×C22).46C22, (C2×Dic11).43C22, (C22×D11).28C22, C11⋊1(C2×C4○D4), (C2×C4×D11)⋊15C2, (C2×C11⋊D4)⋊12C2, SmallGroup(352,176)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D44⋊5C2
G = < a,b,c,d | a2=b44=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd=b22c >
Subgroups: 858 in 164 conjugacy classes, 89 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C11, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, D11, C22, C22, C22, C2×C4○D4, Dic11, C44, D22, D22, C2×C22, C2×C22, C2×C22, Dic22, C4×D11, D44, C2×Dic11, C11⋊D4, C2×C44, C2×C44, C22×D11, C22×C22, C2×Dic22, C2×C4×D11, C2×D44, D44⋊5C2, C2×C11⋊D4, C22×C44, C2×D44⋊5C2
Quotients: C1, C2, C22, C23, C4○D4, C24, D11, C2×C4○D4, D22, C22×D11, D44⋊5C2, C23×D11, C2×D44⋊5C2
►On 176 points
Generators in S176
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(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 45)(42 46)(43 47)(44 48)(89 147)(90 148)(91 149)(92 150)(93 151)(94 152)(95 153)(96 154)(97 155)(98 156)(99 157)(100 158)(101 159)(102 160)(103 161)(104 162)(105 163)(106 164)(107 165)(108 166)(109 167)(110 168)(111 169)(112 170)(113 171)(114 172)(115 173)(116 174)(117 175)(118 176)(119 133)(120 134)(121 135)(122 136)(123 137)(124 138)(125 139)(126 140)(127 141)(128 142)(129 143)(130 144)(131 145)(132 146)
(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 48)(2 47)(3 46)(4 45)(5 88)(6 87)(7 86)(8 85)(9 84)(10 83)(11 82)(12 81)(13 80)(14 79)(15 78)(16 77)(17 76)(18 75)(19 74)(20 73)(21 72)(22 71)(23 70)(24 69)(25 68)(26 67)(27 66)(28 65)(29 64)(30 63)(31 62)(32 61)(33 60)(34 59)(35 58)(36 57)(37 56)(38 55)(39 54)(40 53)(41 52)(42 51)(43 50)(44 49)(89 166)(90 165)(91 164)(92 163)(93 162)(94 161)(95 160)(96 159)(97 158)(98 157)(99 156)(100 155)(101 154)(102 153)(103 152)(104 151)(105 150)(106 149)(107 148)(108 147)(109 146)(110 145)(111 144)(112 143)(113 142)(114 141)(115 140)(116 139)(117 138)(118 137)(119 136)(120 135)(121 134)(122 133)(123 176)(124 175)(125 174)(126 173)(127 172)(128 171)(129 170)(130 169)(131 168)(132 167)
(1 168)(2 169)(3 170)(4 171)(5 172)(6 173)(7 174)(8 175)(9 176)(10 133)(11 134)(12 135)(13 136)(14 137)(15 138)(16 139)(17 140)(18 141)(19 142)(20 143)(21 144)(22 145)(23 146)(24 147)(25 148)(26 149)(27 150)(28 151)(29 152)(30 153)(31 154)(32 155)(33 156)(34 157)(35 158)(36 159)(37 160)(38 161)(39 162)(40 163)(41 164)(42 165)(43 166)(44 167)(45 106)(46 107)(47 108)(48 109)(49 110)(50 111)(51 112)(52 113)(53 114)(54 115)(55 116)(56 117)(57 118)(58 119)(59 120)(60 121)(61 122)(62 123)(63 124)(64 125)(65 126)(66 127)(67 128)(68 129)(69 130)(70 131)(71 132)(72 89)(73 90)(74 91)(75 92)(76 93)(77 94)(78 95)(79 96)(80 97)(81 98)(82 99)(83 100)(84 101)(85 102)(86 103)(87 104)(88 105)
G:=sub<Sym(176)| (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(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)(41,45)(42,46)(43,47)(44,48)(89,147)(90,148)(91,149)(92,150)(93,151)(94,152)(95,153)(96,154)(97,155)(98,156)(99,157)(100,158)(101,159)(102,160)(103,161)(104,162)(105,163)(106,164)(107,165)(108,166)(109,167)(110,168)(111,169)(112,170)(113,171)(114,172)(115,173)(116,174)(117,175)(118,176)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(127,141)(128,142)(129,143)(130,144)(131,145)(132,146), (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), (1,48)(2,47)(3,46)(4,45)(5,88)(6,87)(7,86)(8,85)(9,84)(10,83)(11,82)(12,81)(13,80)(14,79)(15,78)(16,77)(17,76)(18,75)(19,74)(20,73)(21,72)(22,71)(23,70)(24,69)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(89,166)(90,165)(91,164)(92,163)(93,162)(94,161)(95,160)(96,159)(97,158)(98,157)(99,156)(100,155)(101,154)(102,153)(103,152)(104,151)(105,150)(106,149)(107,148)(108,147)(109,146)(110,145)(111,144)(112,143)(113,142)(114,141)(115,140)(116,139)(117,138)(118,137)(119,136)(120,135)(121,134)(122,133)(123,176)(124,175)(125,174)(126,173)(127,172)(128,171)(129,170)(130,169)(131,168)(132,167), (1,168)(2,169)(3,170)(4,171)(5,172)(6,173)(7,174)(8,175)(9,176)(10,133)(11,134)(12,135)(13,136)(14,137)(15,138)(16,139)(17,140)(18,141)(19,142)(20,143)(21,144)(22,145)(23,146)(24,147)(25,148)(26,149)(27,150)(28,151)(29,152)(30,153)(31,154)(32,155)(33,156)(34,157)(35,158)(36,159)(37,160)(38,161)(39,162)(40,163)(41,164)(42,165)(43,166)(44,167)(45,106)(46,107)(47,108)(48,109)(49,110)(50,111)(51,112)(52,113)(53,114)(54,115)(55,116)(56,117)(57,118)(58,119)(59,120)(60,121)(61,122)(62,123)(63,124)(64,125)(65,126)(66,127)(67,128)(68,129)(69,130)(70,131)(71,132)(72,89)(73,90)(74,91)(75,92)(76,93)(77,94)(78,95)(79,96)(80,97)(81,98)(82,99)(83,100)(84,101)(85,102)(86,103)(87,104)(88,105)>;
G:=Group( (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,57)(10,58)(11,59)(12,60)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(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)(41,45)(42,46)(43,47)(44,48)(89,147)(90,148)(91,149)(92,150)(93,151)(94,152)(95,153)(96,154)(97,155)(98,156)(99,157)(100,158)(101,159)(102,160)(103,161)(104,162)(105,163)(106,164)(107,165)(108,166)(109,167)(110,168)(111,169)(112,170)(113,171)(114,172)(115,173)(116,174)(117,175)(118,176)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(127,141)(128,142)(129,143)(130,144)(131,145)(132,146), (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), (1,48)(2,47)(3,46)(4,45)(5,88)(6,87)(7,86)(8,85)(9,84)(10,83)(11,82)(12,81)(13,80)(14,79)(15,78)(16,77)(17,76)(18,75)(19,74)(20,73)(21,72)(22,71)(23,70)(24,69)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(89,166)(90,165)(91,164)(92,163)(93,162)(94,161)(95,160)(96,159)(97,158)(98,157)(99,156)(100,155)(101,154)(102,153)(103,152)(104,151)(105,150)(106,149)(107,148)(108,147)(109,146)(110,145)(111,144)(112,143)(113,142)(114,141)(115,140)(116,139)(117,138)(118,137)(119,136)(120,135)(121,134)(122,133)(123,176)(124,175)(125,174)(126,173)(127,172)(128,171)(129,170)(130,169)(131,168)(132,167), (1,168)(2,169)(3,170)(4,171)(5,172)(6,173)(7,174)(8,175)(9,176)(10,133)(11,134)(12,135)(13,136)(14,137)(15,138)(16,139)(17,140)(18,141)(19,142)(20,143)(21,144)(22,145)(23,146)(24,147)(25,148)(26,149)(27,150)(28,151)(29,152)(30,153)(31,154)(32,155)(33,156)(34,157)(35,158)(36,159)(37,160)(38,161)(39,162)(40,163)(41,164)(42,165)(43,166)(44,167)(45,106)(46,107)(47,108)(48,109)(49,110)(50,111)(51,112)(52,113)(53,114)(54,115)(55,116)(56,117)(57,118)(58,119)(59,120)(60,121)(61,122)(62,123)(63,124)(64,125)(65,126)(66,127)(67,128)(68,129)(69,130)(70,131)(71,132)(72,89)(73,90)(74,91)(75,92)(76,93)(77,94)(78,95)(79,96)(80,97)(81,98)(82,99)(83,100)(84,101)(85,102)(86,103)(87,104)(88,105) );
G=PermutationGroup([[(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,57),(10,58),(11,59),(12,60),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(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),(41,45),(42,46),(43,47),(44,48),(89,147),(90,148),(91,149),(92,150),(93,151),(94,152),(95,153),(96,154),(97,155),(98,156),(99,157),(100,158),(101,159),(102,160),(103,161),(104,162),(105,163),(106,164),(107,165),(108,166),(109,167),(110,168),(111,169),(112,170),(113,171),(114,172),(115,173),(116,174),(117,175),(118,176),(119,133),(120,134),(121,135),(122,136),(123,137),(124,138),(125,139),(126,140),(127,141),(128,142),(129,143),(130,144),(131,145),(132,146)], [(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)], [(1,48),(2,47),(3,46),(4,45),(5,88),(6,87),(7,86),(8,85),(9,84),(10,83),(11,82),(12,81),(13,80),(14,79),(15,78),(16,77),(17,76),(18,75),(19,74),(20,73),(21,72),(22,71),(23,70),(24,69),(25,68),(26,67),(27,66),(28,65),(29,64),(30,63),(31,62),(32,61),(33,60),(34,59),(35,58),(36,57),(37,56),(38,55),(39,54),(40,53),(41,52),(42,51),(43,50),(44,49),(89,166),(90,165),(91,164),(92,163),(93,162),(94,161),(95,160),(96,159),(97,158),(98,157),(99,156),(100,155),(101,154),(102,153),(103,152),(104,151),(105,150),(106,149),(107,148),(108,147),(109,146),(110,145),(111,144),(112,143),(113,142),(114,141),(115,140),(116,139),(117,138),(118,137),(119,136),(120,135),(121,134),(122,133),(123,176),(124,175),(125,174),(126,173),(127,172),(128,171),(129,170),(130,169),(131,168),(132,167)], [(1,168),(2,169),(3,170),(4,171),(5,172),(6,173),(7,174),(8,175),(9,176),(10,133),(11,134),(12,135),(13,136),(14,137),(15,138),(16,139),(17,140),(18,141),(19,142),(20,143),(21,144),(22,145),(23,146),(24,147),(25,148),(26,149),(27,150),(28,151),(29,152),(30,153),(31,154),(32,155),(33,156),(34,157),(35,158),(36,159),(37,160),(38,161),(39,162),(40,163),(41,164),(42,165),(43,166),(44,167),(45,106),(46,107),(47,108),(48,109),(49,110),(50,111),(51,112),(52,113),(53,114),(54,115),(55,116),(56,117),(57,118),(58,119),(59,120),(60,121),(61,122),(62,123),(63,124),(64,125),(65,126),(66,127),(67,128),(68,129),(69,130),(70,131),(71,132),(72,89),(73,90),(74,91),(75,92),(76,93),(77,94),(78,95),(79,96),(80,97),(81,98),(82,99),(83,100),(84,101),(85,102),(86,103),(87,104),(88,105)]])
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 11A | ··· | 11E | 22A | ··· | 22AI | 44A | ··· | 44AN |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 22 | 22 | 22 | 22 | 1 | 1 | 1 | 1 | 2 | 2 | 22 | 22 | 22 | 22 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4○D4 | D11 | D22 | D22 | D44⋊5C2 |
| kernel | C2×D44⋊5C2 | C2×Dic22 | C2×C4×D11 | C2×D44 | D44⋊5C2 | C2×C11⋊D4 | C22×C44 | C22 | C22×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 1 | 2 | 1 | 8 | 2 | 1 | 4 | 5 | 30 | 5 | 40 |
Matrix representation of C2×D44⋊5C2 ►in GL3(𝔽89) generated by
| 88 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 44 | 16 |
| 0 | 73 | 65 |
| 1 | 0 | 0 |
| 0 | 65 | 3 |
| 0 | 16 | 24 |
| 1 | 0 | 0 |
| 0 | 76 | 41 |
| 0 | 48 | 13 |
G:=sub<GL(3,GF(89))| [88,0,0,0,1,0,0,0,1],[1,0,0,0,44,73,0,16,65],[1,0,0,0,65,16,0,3,24],[1,0,0,0,76,48,0,41,13] >;
C2×D44⋊5C2 in GAP, Magma, Sage, TeX
C_2\times D_{44}\rtimes_5C_2 % in TeX
G:=Group("C2xD44:5C2"); // GroupNames label
G:=SmallGroup(352,176);
// by ID
G=gap.SmallGroup(352,176);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-11,86,579,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^44=c^2=d^2=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=b^22*c>;
// generators/relations