metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C4).22D20, (C2×C20).33D4, (C22×D5).2Q8, C22.44(Q8×D5), C2.4(C4⋊D20), C10.2(C4⋊1D4), (C22×C4).73D10, C22.84(C2×D20), C5⋊1(C23.4Q8), C2.9(D10⋊2Q8), C2.C42⋊14D5, C10.28(C22⋊Q8), (C23×D5).9C22, (C22×C20).47C22, C23.365(C22×D5), C22.91(D4⋊2D5), (C22×C10).302C23, C2.9(C22.D20), C10.13(C22.D4), (C22×Dic5).24C22, (C2×C4⋊Dic5)⋊4C2, (C2×C10).98(C2×D4), (C2×C10).71(C2×Q8), (C2×D10⋊C4).19C2, (C2×C10).136(C4○D4), (C5×C2.C42)⋊12C2, SmallGroup(320,304)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C4).22D20
G = < a,b,c,d | a2=b4=c20=1, d2=a, cbc-1=ab=ba, ac=ca, ad=da, dbd-1=b-1, dcd-1=ac-1 >
Subgroups: 790 in 186 conjugacy classes, 65 normal (12 characteristic)
C1, C2, C2 [×6], C2 [×2], C4 [×9], C22, C22 [×6], C22 [×10], C5, C2×C4 [×6], C2×C4 [×15], C23, C23 [×8], D5 [×2], C10, C10 [×6], C22⋊C4 [×6], C4⋊C4 [×6], C22×C4 [×3], C22×C4 [×3], C24, Dic5 [×3], C20 [×6], D10 [×10], C2×C10, C2×C10 [×6], C2.C42, C2×C22⋊C4 [×3], C2×C4⋊C4 [×3], C2×Dic5 [×9], C2×C20 [×6], C2×C20 [×6], C22×D5 [×2], C22×D5 [×6], C22×C10, C23.4Q8, C4⋊Dic5 [×6], D10⋊C4 [×6], C22×Dic5 [×3], C22×C20 [×3], C23×D5, C5×C2.C42, C2×C4⋊Dic5 [×3], C2×D10⋊C4 [×3], (C2×C4).22D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, D5, C2×D4 [×3], C2×Q8, C4○D4 [×3], D10 [×3], C22⋊Q8 [×3], C22.D4 [×3], C4⋊1D4, D20 [×6], C22×D5, C23.4Q8, C2×D20 [×3], D4⋊2D5 [×3], Q8×D5, C4⋊D20, C22.D20 [×3], D10⋊2Q8 [×3], (C2×C4).22D20
►On 160 points
Generators in S160
(1 114)(2 115)(3 116)(4 117)(5 118)(6 119)(7 120)(8 101)(9 102)(10 103)(11 104)(12 105)(13 106)(14 107)(15 108)(16 109)(17 110)(18 111)(19 112)(20 113)(21 43)(22 44)(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 41)(40 42)(61 150)(62 151)(63 152)(64 153)(65 154)(66 155)(67 156)(68 157)(69 158)(70 159)(71 160)(72 141)(73 142)(74 143)(75 144)(76 145)(77 146)(78 147)(79 148)(80 149)(81 139)(82 140)(83 121)(84 122)(85 123)(86 124)(87 125)(88 126)(89 127)(90 128)(91 129)(92 130)(93 131)(94 132)(95 133)(96 134)(97 135)(98 136)(99 137)(100 138)
(1 71 42 86)(2 141 43 125)(3 73 44 88)(4 143 45 127)(5 75 46 90)(6 145 47 129)(7 77 48 92)(8 147 49 131)(9 79 50 94)(10 149 51 133)(11 61 52 96)(12 151 53 135)(13 63 54 98)(14 153 55 137)(15 65 56 100)(16 155 57 139)(17 67 58 82)(18 157 59 121)(19 69 60 84)(20 159 41 123)(21 87 115 72)(22 126 116 142)(23 89 117 74)(24 128 118 144)(25 91 119 76)(26 130 120 146)(27 93 101 78)(28 132 102 148)(29 95 103 80)(30 134 104 150)(31 97 105 62)(32 136 106 152)(33 99 107 64)(34 138 108 154)(35 81 109 66)(36 140 110 156)(37 83 111 68)(38 122 112 158)(39 85 113 70)(40 124 114 160)
(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)
(1 113 114 20)(2 19 115 112)(3 111 116 18)(4 17 117 110)(5 109 118 16)(6 15 119 108)(7 107 120 14)(8 13 101 106)(9 105 102 12)(10 11 103 104)(21 38 43 60)(22 59 44 37)(23 36 45 58)(24 57 46 35)(25 34 47 56)(26 55 48 33)(27 32 49 54)(28 53 50 31)(29 30 51 52)(39 40 41 42)(61 95 150 133)(62 132 151 94)(63 93 152 131)(64 130 153 92)(65 91 154 129)(66 128 155 90)(67 89 156 127)(68 126 157 88)(69 87 158 125)(70 124 159 86)(71 85 160 123)(72 122 141 84)(73 83 142 121)(74 140 143 82)(75 81 144 139)(76 138 145 100)(77 99 146 137)(78 136 147 98)(79 97 148 135)(80 134 149 96)
G:=sub<Sym(160)| (1,114)(2,115)(3,116)(4,117)(5,118)(6,119)(7,120)(8,101)(9,102)(10,103)(11,104)(12,105)(13,106)(14,107)(15,108)(16,109)(17,110)(18,111)(19,112)(20,113)(21,43)(22,44)(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)(39,41)(40,42)(61,150)(62,151)(63,152)(64,153)(65,154)(66,155)(67,156)(68,157)(69,158)(70,159)(71,160)(72,141)(73,142)(74,143)(75,144)(76,145)(77,146)(78,147)(79,148)(80,149)(81,139)(82,140)(83,121)(84,122)(85,123)(86,124)(87,125)(88,126)(89,127)(90,128)(91,129)(92,130)(93,131)(94,132)(95,133)(96,134)(97,135)(98,136)(99,137)(100,138), (1,71,42,86)(2,141,43,125)(3,73,44,88)(4,143,45,127)(5,75,46,90)(6,145,47,129)(7,77,48,92)(8,147,49,131)(9,79,50,94)(10,149,51,133)(11,61,52,96)(12,151,53,135)(13,63,54,98)(14,153,55,137)(15,65,56,100)(16,155,57,139)(17,67,58,82)(18,157,59,121)(19,69,60,84)(20,159,41,123)(21,87,115,72)(22,126,116,142)(23,89,117,74)(24,128,118,144)(25,91,119,76)(26,130,120,146)(27,93,101,78)(28,132,102,148)(29,95,103,80)(30,134,104,150)(31,97,105,62)(32,136,106,152)(33,99,107,64)(34,138,108,154)(35,81,109,66)(36,140,110,156)(37,83,111,68)(38,122,112,158)(39,85,113,70)(40,124,114,160), (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), (1,113,114,20)(2,19,115,112)(3,111,116,18)(4,17,117,110)(5,109,118,16)(6,15,119,108)(7,107,120,14)(8,13,101,106)(9,105,102,12)(10,11,103,104)(21,38,43,60)(22,59,44,37)(23,36,45,58)(24,57,46,35)(25,34,47,56)(26,55,48,33)(27,32,49,54)(28,53,50,31)(29,30,51,52)(39,40,41,42)(61,95,150,133)(62,132,151,94)(63,93,152,131)(64,130,153,92)(65,91,154,129)(66,128,155,90)(67,89,156,127)(68,126,157,88)(69,87,158,125)(70,124,159,86)(71,85,160,123)(72,122,141,84)(73,83,142,121)(74,140,143,82)(75,81,144,139)(76,138,145,100)(77,99,146,137)(78,136,147,98)(79,97,148,135)(80,134,149,96)>;
G:=Group( (1,114)(2,115)(3,116)(4,117)(5,118)(6,119)(7,120)(8,101)(9,102)(10,103)(11,104)(12,105)(13,106)(14,107)(15,108)(16,109)(17,110)(18,111)(19,112)(20,113)(21,43)(22,44)(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)(39,41)(40,42)(61,150)(62,151)(63,152)(64,153)(65,154)(66,155)(67,156)(68,157)(69,158)(70,159)(71,160)(72,141)(73,142)(74,143)(75,144)(76,145)(77,146)(78,147)(79,148)(80,149)(81,139)(82,140)(83,121)(84,122)(85,123)(86,124)(87,125)(88,126)(89,127)(90,128)(91,129)(92,130)(93,131)(94,132)(95,133)(96,134)(97,135)(98,136)(99,137)(100,138), (1,71,42,86)(2,141,43,125)(3,73,44,88)(4,143,45,127)(5,75,46,90)(6,145,47,129)(7,77,48,92)(8,147,49,131)(9,79,50,94)(10,149,51,133)(11,61,52,96)(12,151,53,135)(13,63,54,98)(14,153,55,137)(15,65,56,100)(16,155,57,139)(17,67,58,82)(18,157,59,121)(19,69,60,84)(20,159,41,123)(21,87,115,72)(22,126,116,142)(23,89,117,74)(24,128,118,144)(25,91,119,76)(26,130,120,146)(27,93,101,78)(28,132,102,148)(29,95,103,80)(30,134,104,150)(31,97,105,62)(32,136,106,152)(33,99,107,64)(34,138,108,154)(35,81,109,66)(36,140,110,156)(37,83,111,68)(38,122,112,158)(39,85,113,70)(40,124,114,160), (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), (1,113,114,20)(2,19,115,112)(3,111,116,18)(4,17,117,110)(5,109,118,16)(6,15,119,108)(7,107,120,14)(8,13,101,106)(9,105,102,12)(10,11,103,104)(21,38,43,60)(22,59,44,37)(23,36,45,58)(24,57,46,35)(25,34,47,56)(26,55,48,33)(27,32,49,54)(28,53,50,31)(29,30,51,52)(39,40,41,42)(61,95,150,133)(62,132,151,94)(63,93,152,131)(64,130,153,92)(65,91,154,129)(66,128,155,90)(67,89,156,127)(68,126,157,88)(69,87,158,125)(70,124,159,86)(71,85,160,123)(72,122,141,84)(73,83,142,121)(74,140,143,82)(75,81,144,139)(76,138,145,100)(77,99,146,137)(78,136,147,98)(79,97,148,135)(80,134,149,96) );
G=PermutationGroup([(1,114),(2,115),(3,116),(4,117),(5,118),(6,119),(7,120),(8,101),(9,102),(10,103),(11,104),(12,105),(13,106),(14,107),(15,108),(16,109),(17,110),(18,111),(19,112),(20,113),(21,43),(22,44),(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),(39,41),(40,42),(61,150),(62,151),(63,152),(64,153),(65,154),(66,155),(67,156),(68,157),(69,158),(70,159),(71,160),(72,141),(73,142),(74,143),(75,144),(76,145),(77,146),(78,147),(79,148),(80,149),(81,139),(82,140),(83,121),(84,122),(85,123),(86,124),(87,125),(88,126),(89,127),(90,128),(91,129),(92,130),(93,131),(94,132),(95,133),(96,134),(97,135),(98,136),(99,137),(100,138)], [(1,71,42,86),(2,141,43,125),(3,73,44,88),(4,143,45,127),(5,75,46,90),(6,145,47,129),(7,77,48,92),(8,147,49,131),(9,79,50,94),(10,149,51,133),(11,61,52,96),(12,151,53,135),(13,63,54,98),(14,153,55,137),(15,65,56,100),(16,155,57,139),(17,67,58,82),(18,157,59,121),(19,69,60,84),(20,159,41,123),(21,87,115,72),(22,126,116,142),(23,89,117,74),(24,128,118,144),(25,91,119,76),(26,130,120,146),(27,93,101,78),(28,132,102,148),(29,95,103,80),(30,134,104,150),(31,97,105,62),(32,136,106,152),(33,99,107,64),(34,138,108,154),(35,81,109,66),(36,140,110,156),(37,83,111,68),(38,122,112,158),(39,85,113,70),(40,124,114,160)], [(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)], [(1,113,114,20),(2,19,115,112),(3,111,116,18),(4,17,117,110),(5,109,118,16),(6,15,119,108),(7,107,120,14),(8,13,101,106),(9,105,102,12),(10,11,103,104),(21,38,43,60),(22,59,44,37),(23,36,45,58),(24,57,46,35),(25,34,47,56),(26,55,48,33),(27,32,49,54),(28,53,50,31),(29,30,51,52),(39,40,41,42),(61,95,150,133),(62,132,151,94),(63,93,152,131),(64,130,153,92),(65,91,154,129),(66,128,155,90),(67,89,156,127),(68,126,157,88),(69,87,158,125),(70,124,159,86),(71,85,160,123),(72,122,141,84),(73,83,142,121),(74,140,143,82),(75,81,144,139),(76,138,145,100),(77,99,146,137),(78,136,147,98),(79,97,148,135),(80,134,149,96)])
62 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | ··· | 4F | 4G | ··· | 4L | 5A | 5B | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 20 | 20 | 4 | ··· | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | + | - | - | |
| image | C1 | C2 | C2 | C2 | D4 | Q8 | D5 | C4○D4 | D10 | D20 | D4⋊2D5 | Q8×D5 |
| kernel | (C2×C4).22D20 | C5×C2.C42 | C2×C4⋊Dic5 | C2×D10⋊C4 | C2×C20 | C22×D5 | C2.C42 | C2×C10 | C22×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 3 | 3 | 6 | 2 | 2 | 6 | 6 | 24 | 6 | 2 |
Matrix representation of (C2×C4).22D20 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 30 | 13 | 0 | 0 | 0 | 0 |
| 19 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 9 | 0 | 0 |
| 0 | 0 | 4 | 39 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 40 |
| 0 | 0 | 0 | 0 | 39 | 32 |
| 1 | 40 | 0 | 0 | 0 | 0 |
| 8 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 30 | 0 | 0 |
| 0 | 0 | 27 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 9 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 33 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 39 | 32 | 0 | 0 |
| 0 | 0 | 14 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 9 |
| 0 | 0 | 0 | 0 | 18 | 40 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[30,19,0,0,0,0,13,11,0,0,0,0,0,0,2,4,0,0,0,0,9,39,0,0,0,0,0,0,9,39,0,0,0,0,40,32],[1,8,0,0,0,0,40,34,0,0,0,0,0,0,2,27,0,0,0,0,30,16,0,0,0,0,0,0,1,0,0,0,0,0,9,40],[40,33,0,0,0,0,0,1,0,0,0,0,0,0,39,14,0,0,0,0,32,2,0,0,0,0,0,0,1,18,0,0,0,0,9,40] >;
(C2×C4).22D20 in GAP, Magma, Sage, TeX
(C_2\times C_4)._{22}D_{20} % in TeX
G:=Group("(C2xC4).22D20"); // GroupNames label
G:=SmallGroup(320,304);
// by ID
G=gap.SmallGroup(320,304);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,387,226,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^20=1,d^2=a,c*b*c^-1=a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b^-1,d*c*d^-1=a*c^-1>;
// generators/relations