direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22xDic10, C10.1C24, C20.34C23, C23.33D10, Dic5.1C23, (C2xC10):4Q8, C10:1(C2xQ8), C5:1(C22xQ8), (C2xC4).86D10, (C22xC4).8D5, C2.3(C23xD5), (C22xC20).8C2, C4.32(C22xD5), (C2xC20).95C22, (C2xC10).62C23, (C22xDic5).6C2, C22.28(C22xD5), (C22xC10).43C22, (C2xDic5).45C22, SmallGroup(160,213)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22xDic10
G = < a,b,c,d | a2=b2=c20=1, d2=c10, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 360 in 156 conjugacy classes, 105 normal (9 characteristic)
C1, C2, C2, C4, C4, C22, C5, C2xC4, C2xC4, Q8, C23, C10, C10, C22xC4, C22xC4, C2xQ8, Dic5, C20, C2xC10, C22xQ8, Dic10, C2xDic5, C2xC20, C22xC10, C2xDic10, C22xDic5, C22xC20, C22xDic10
Quotients: C1, C2, C22, Q8, C23, D5, C2xQ8, C24, D10, C22xQ8, Dic10, C22xD5, C2xDic10, C23xD5, C22xDic10
►Regular action on 160 points
Generators in S160
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 57)(16 58)(17 59)(18 60)(19 41)(20 42)(21 121)(22 122)(23 123)(24 124)(25 125)(26 126)(27 127)(28 128)(29 129)(30 130)(31 131)(32 132)(33 133)(34 134)(35 135)(36 136)(37 137)(38 138)(39 139)(40 140)(61 113)(62 114)(63 115)(64 116)(65 117)(66 118)(67 119)(68 120)(69 101)(70 102)(71 103)(72 104)(73 105)(74 106)(75 107)(76 108)(77 109)(78 110)(79 111)(80 112)(81 160)(82 141)(83 142)(84 143)(85 144)(86 145)(87 146)(88 147)(89 148)(90 149)(91 150)(92 151)(93 152)(94 153)(95 154)(96 155)(97 156)(98 157)(99 158)(100 159)
(1 140)(2 121)(3 122)(4 123)(5 124)(6 125)(7 126)(8 127)(9 128)(10 129)(11 130)(12 131)(13 132)(14 133)(15 134)(16 135)(17 136)(18 137)(19 138)(20 139)(21 44)(22 45)(23 46)(24 47)(25 48)(26 49)(27 50)(28 51)(29 52)(30 53)(31 54)(32 55)(33 56)(34 57)(35 58)(36 59)(37 60)(38 41)(39 42)(40 43)(61 155)(62 156)(63 157)(64 158)(65 159)(66 160)(67 141)(68 142)(69 143)(70 144)(71 145)(72 146)(73 147)(74 148)(75 149)(76 150)(77 151)(78 152)(79 153)(80 154)(81 118)(82 119)(83 120)(84 101)(85 102)(86 103)(87 104)(88 105)(89 106)(90 107)(91 108)(92 109)(93 110)(94 111)(95 112)(96 113)(97 114)(98 115)(99 116)(100 117)
(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)
(1 106 11 116)(2 105 12 115)(3 104 13 114)(4 103 14 113)(5 102 15 112)(6 101 16 111)(7 120 17 110)(8 119 18 109)(9 118 19 108)(10 117 20 107)(21 147 31 157)(22 146 32 156)(23 145 33 155)(24 144 34 154)(25 143 35 153)(26 142 36 152)(27 141 37 151)(28 160 38 150)(29 159 39 149)(30 158 40 148)(41 76 51 66)(42 75 52 65)(43 74 53 64)(44 73 54 63)(45 72 55 62)(46 71 56 61)(47 70 57 80)(48 69 58 79)(49 68 59 78)(50 67 60 77)(81 138 91 128)(82 137 92 127)(83 136 93 126)(84 135 94 125)(85 134 95 124)(86 133 96 123)(87 132 97 122)(88 131 98 121)(89 130 99 140)(90 129 100 139)
G:=sub<Sym(160)| (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,41)(20,42)(21,121)(22,122)(23,123)(24,124)(25,125)(26,126)(27,127)(28,128)(29,129)(30,130)(31,131)(32,132)(33,133)(34,134)(35,135)(36,136)(37,137)(38,138)(39,139)(40,140)(61,113)(62,114)(63,115)(64,116)(65,117)(66,118)(67,119)(68,120)(69,101)(70,102)(71,103)(72,104)(73,105)(74,106)(75,107)(76,108)(77,109)(78,110)(79,111)(80,112)(81,160)(82,141)(83,142)(84,143)(85,144)(86,145)(87,146)(88,147)(89,148)(90,149)(91,150)(92,151)(93,152)(94,153)(95,154)(96,155)(97,156)(98,157)(99,158)(100,159), (1,140)(2,121)(3,122)(4,123)(5,124)(6,125)(7,126)(8,127)(9,128)(10,129)(11,130)(12,131)(13,132)(14,133)(15,134)(16,135)(17,136)(18,137)(19,138)(20,139)(21,44)(22,45)(23,46)(24,47)(25,48)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59)(37,60)(38,41)(39,42)(40,43)(61,155)(62,156)(63,157)(64,158)(65,159)(66,160)(67,141)(68,142)(69,143)(70,144)(71,145)(72,146)(73,147)(74,148)(75,149)(76,150)(77,151)(78,152)(79,153)(80,154)(81,118)(82,119)(83,120)(84,101)(85,102)(86,103)(87,104)(88,105)(89,106)(90,107)(91,108)(92,109)(93,110)(94,111)(95,112)(96,113)(97,114)(98,115)(99,116)(100,117), (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), (1,106,11,116)(2,105,12,115)(3,104,13,114)(4,103,14,113)(5,102,15,112)(6,101,16,111)(7,120,17,110)(8,119,18,109)(9,118,19,108)(10,117,20,107)(21,147,31,157)(22,146,32,156)(23,145,33,155)(24,144,34,154)(25,143,35,153)(26,142,36,152)(27,141,37,151)(28,160,38,150)(29,159,39,149)(30,158,40,148)(41,76,51,66)(42,75,52,65)(43,74,53,64)(44,73,54,63)(45,72,55,62)(46,71,56,61)(47,70,57,80)(48,69,58,79)(49,68,59,78)(50,67,60,77)(81,138,91,128)(82,137,92,127)(83,136,93,126)(84,135,94,125)(85,134,95,124)(86,133,96,123)(87,132,97,122)(88,131,98,121)(89,130,99,140)(90,129,100,139)>;
G:=Group( (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,49)(8,50)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,41)(20,42)(21,121)(22,122)(23,123)(24,124)(25,125)(26,126)(27,127)(28,128)(29,129)(30,130)(31,131)(32,132)(33,133)(34,134)(35,135)(36,136)(37,137)(38,138)(39,139)(40,140)(61,113)(62,114)(63,115)(64,116)(65,117)(66,118)(67,119)(68,120)(69,101)(70,102)(71,103)(72,104)(73,105)(74,106)(75,107)(76,108)(77,109)(78,110)(79,111)(80,112)(81,160)(82,141)(83,142)(84,143)(85,144)(86,145)(87,146)(88,147)(89,148)(90,149)(91,150)(92,151)(93,152)(94,153)(95,154)(96,155)(97,156)(98,157)(99,158)(100,159), (1,140)(2,121)(3,122)(4,123)(5,124)(6,125)(7,126)(8,127)(9,128)(10,129)(11,130)(12,131)(13,132)(14,133)(15,134)(16,135)(17,136)(18,137)(19,138)(20,139)(21,44)(22,45)(23,46)(24,47)(25,48)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59)(37,60)(38,41)(39,42)(40,43)(61,155)(62,156)(63,157)(64,158)(65,159)(66,160)(67,141)(68,142)(69,143)(70,144)(71,145)(72,146)(73,147)(74,148)(75,149)(76,150)(77,151)(78,152)(79,153)(80,154)(81,118)(82,119)(83,120)(84,101)(85,102)(86,103)(87,104)(88,105)(89,106)(90,107)(91,108)(92,109)(93,110)(94,111)(95,112)(96,113)(97,114)(98,115)(99,116)(100,117), (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), (1,106,11,116)(2,105,12,115)(3,104,13,114)(4,103,14,113)(5,102,15,112)(6,101,16,111)(7,120,17,110)(8,119,18,109)(9,118,19,108)(10,117,20,107)(21,147,31,157)(22,146,32,156)(23,145,33,155)(24,144,34,154)(25,143,35,153)(26,142,36,152)(27,141,37,151)(28,160,38,150)(29,159,39,149)(30,158,40,148)(41,76,51,66)(42,75,52,65)(43,74,53,64)(44,73,54,63)(45,72,55,62)(46,71,56,61)(47,70,57,80)(48,69,58,79)(49,68,59,78)(50,67,60,77)(81,138,91,128)(82,137,92,127)(83,136,93,126)(84,135,94,125)(85,134,95,124)(86,133,96,123)(87,132,97,122)(88,131,98,121)(89,130,99,140)(90,129,100,139) );
G=PermutationGroup([[(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,49),(8,50),(9,51),(10,52),(11,53),(12,54),(13,55),(14,56),(15,57),(16,58),(17,59),(18,60),(19,41),(20,42),(21,121),(22,122),(23,123),(24,124),(25,125),(26,126),(27,127),(28,128),(29,129),(30,130),(31,131),(32,132),(33,133),(34,134),(35,135),(36,136),(37,137),(38,138),(39,139),(40,140),(61,113),(62,114),(63,115),(64,116),(65,117),(66,118),(67,119),(68,120),(69,101),(70,102),(71,103),(72,104),(73,105),(74,106),(75,107),(76,108),(77,109),(78,110),(79,111),(80,112),(81,160),(82,141),(83,142),(84,143),(85,144),(86,145),(87,146),(88,147),(89,148),(90,149),(91,150),(92,151),(93,152),(94,153),(95,154),(96,155),(97,156),(98,157),(99,158),(100,159)], [(1,140),(2,121),(3,122),(4,123),(5,124),(6,125),(7,126),(8,127),(9,128),(10,129),(11,130),(12,131),(13,132),(14,133),(15,134),(16,135),(17,136),(18,137),(19,138),(20,139),(21,44),(22,45),(23,46),(24,47),(25,48),(26,49),(27,50),(28,51),(29,52),(30,53),(31,54),(32,55),(33,56),(34,57),(35,58),(36,59),(37,60),(38,41),(39,42),(40,43),(61,155),(62,156),(63,157),(64,158),(65,159),(66,160),(67,141),(68,142),(69,143),(70,144),(71,145),(72,146),(73,147),(74,148),(75,149),(76,150),(77,151),(78,152),(79,153),(80,154),(81,118),(82,119),(83,120),(84,101),(85,102),(86,103),(87,104),(88,105),(89,106),(90,107),(91,108),(92,109),(93,110),(94,111),(95,112),(96,113),(97,114),(98,115),(99,116),(100,117)], [(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)], [(1,106,11,116),(2,105,12,115),(3,104,13,114),(4,103,14,113),(5,102,15,112),(6,101,16,111),(7,120,17,110),(8,119,18,109),(9,118,19,108),(10,117,20,107),(21,147,31,157),(22,146,32,156),(23,145,33,155),(24,144,34,154),(25,143,35,153),(26,142,36,152),(27,141,37,151),(28,160,38,150),(29,159,39,149),(30,158,40,148),(41,76,51,66),(42,75,52,65),(43,74,53,64),(44,73,54,63),(45,72,55,62),(46,71,56,61),(47,70,57,80),(48,69,58,79),(49,68,59,78),(50,67,60,77),(81,138,91,128),(82,137,92,127),(83,136,93,126),(84,135,94,125),(85,134,95,124),(86,133,96,123),(87,132,97,122),(88,131,98,121),(89,130,99,140),(90,129,100,139)]])
C22xDic10 is a maximal subgroup of
(C2xC20):Q8 (C2xDic5):Q8 (C2xC4).20D20 Dic10:14D4 C22:Dic20 (C2xC20):10Q8 C23:Dic10 (C2xDic5):6Q8 (C2xC4):Dic10 C4.(C2xD20) Dic10:17D4 Dic10.37D4 C23.46D20 C42.87D10 C42.92D10 Dic10:23D4 Dic10:19D4 Dic10:21D4 C10.792- 1+4 C10.1052- 1+4 C22xQ8xD5
C22xDic10 is a maximal quotient of
C42.274D10 C23:2Dic10 C10.12- 1+4 C42.88D10 C42.90D10 D4:5Dic10 D4:6Dic10 Q8:5Dic10 Q8:6Dic10
52 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20P |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | + | - |
| image | C1 | C2 | C2 | C2 | Q8 | D5 | D10 | D10 | Dic10 |
| kernel | C22xDic10 | C2xDic10 | C22xDic5 | C22xC20 | C2xC10 | C22xC4 | C2xC4 | C23 | C22 |
| # reps | 1 | 12 | 2 | 1 | 4 | 2 | 12 | 2 | 16 |
Matrix representation of C22xDic10 ►in GL4(F41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 39 |
| 0 | 0 | 2 | 16 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 28 | 9 |
| 0 | 0 | 13 | 13 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,1,0,0,0,0,28,2,0,0,39,16],[40,0,0,0,0,40,0,0,0,0,28,13,0,0,9,13] >;
C22xDic10 in GAP, Magma, Sage, TeX
C_2^2\times {\rm Dic}_{10} % in TeX
G:=Group("C2^2xDic10"); // GroupNames label
G:=SmallGroup(160,213);
// by ID
G=gap.SmallGroup(160,213);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,579,69,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^20=1,d^2=c^10,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations