direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C4.12D20, M4(2).32D10, (C2×C4).53D20, C4.67(C2×D20), C20.422(C2×D4), (C2×C20).176D4, C23.56(C4×D5), C10⋊2(C4.10D4), (C2×C20).418C23, (C2×Dic10).28C4, (C22×C4).144D10, (C2×M4(2)).17D5, C4.29(D10⋊C4), (C22×Dic5).5C4, C20.101(C22⋊C4), (C10×M4(2)).28C2, C4.Dic5.43C22, (C22×C20).191C22, (C22×Dic10).16C2, (C5×M4(2)).35C22, C22.51(D10⋊C4), (C2×Dic10).287C22, (C2×C4).55(C4×D5), C5⋊4(C2×C4.10D4), C22.22(C2×C4×D5), C4.115(C2×C5⋊D4), (C2×C20).284(C2×C4), (C2×Dic5).6(C2×C4), C2.33(C2×D10⋊C4), (C2×C4).257(C5⋊D4), C10.102(C2×C22⋊C4), (C2×C4).122(C22×D5), (C2×C4.Dic5).26C2, (C2×C10).117(C22×C4), (C22×C10).140(C2×C4), (C2×C10).131(C22⋊C4), SmallGroup(320,763)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C4.12D20
G = < a,b,c,d | a2=b20=1, c4=d2=b10, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b5c3 >
Subgroups: 478 in 146 conjugacy classes, 63 normal (39 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, Q8, C23, C10, C10, C10, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×Q8, Dic5, C20, C2×C10, C2×C10, C4.10D4, C2×M4(2), C2×M4(2), C22×Q8, C5⋊2C8, C40, Dic10, C2×Dic5, C2×Dic5, C2×C20, C22×C10, C2×C4.10D4, C2×C5⋊2C8, C4.Dic5, C4.Dic5, C2×C40, C5×M4(2), C5×M4(2), C2×Dic10, C2×Dic10, C22×Dic5, C22×C20, C4.12D20, C2×C4.Dic5, C10×M4(2), C22×Dic10, C2×C4.12D20
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, D10, C4.10D4, C2×C22⋊C4, C4×D5, D20, C5⋊D4, C22×D5, C2×C4.10D4, D10⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, C4.12D20, C2×D10⋊C4, C2×C4.12D20
►On 160 points
Generators in S160
(1 122)(2 123)(3 124)(4 125)(5 126)(6 127)(7 128)(8 129)(9 130)(10 131)(11 132)(12 133)(13 134)(14 135)(15 136)(16 137)(17 138)(18 139)(19 140)(20 121)(21 99)(22 100)(23 81)(24 82)(25 83)(26 84)(27 85)(28 86)(29 87)(30 88)(31 89)(32 90)(33 91)(34 92)(35 93)(36 94)(37 95)(38 96)(39 97)(40 98)(41 150)(42 151)(43 152)(44 153)(45 154)(46 155)(47 156)(48 157)(49 158)(50 159)(51 160)(52 141)(53 142)(54 143)(55 144)(56 145)(57 146)(58 147)(59 148)(60 149)(61 112)(62 113)(63 114)(64 115)(65 116)(66 117)(67 118)(68 119)(69 120)(70 101)(71 102)(72 103)(73 104)(74 105)(75 106)(76 107)(77 108)(78 109)(79 110)(80 111)
(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)
(1 159 127 45 11 149 137 55)(2 158 128 44 12 148 138 54)(3 157 129 43 13 147 139 53)(4 156 130 42 14 146 140 52)(5 155 131 41 15 145 121 51)(6 154 132 60 16 144 122 50)(7 153 133 59 17 143 123 49)(8 152 134 58 18 142 124 48)(9 151 135 57 19 141 125 47)(10 150 136 56 20 160 126 46)(21 105 94 79 31 115 84 69)(22 104 95 78 32 114 85 68)(23 103 96 77 33 113 86 67)(24 102 97 76 34 112 87 66)(25 101 98 75 35 111 88 65)(26 120 99 74 36 110 89 64)(27 119 100 73 37 109 90 63)(28 118 81 72 38 108 91 62)(29 117 82 71 39 107 92 61)(30 116 83 70 40 106 93 80)
(1 102 11 112)(2 101 12 111)(3 120 13 110)(4 119 14 109)(5 118 15 108)(6 117 16 107)(7 116 17 106)(8 115 18 105)(9 114 19 104)(10 113 20 103)(21 157 31 147)(22 156 32 146)(23 155 33 145)(24 154 34 144)(25 153 35 143)(26 152 36 142)(27 151 37 141)(28 150 38 160)(29 149 39 159)(30 148 40 158)(41 96 51 86)(42 95 52 85)(43 94 53 84)(44 93 54 83)(45 92 55 82)(46 91 56 81)(47 90 57 100)(48 89 58 99)(49 88 59 98)(50 87 60 97)(61 122 71 132)(62 121 72 131)(63 140 73 130)(64 139 74 129)(65 138 75 128)(66 137 76 127)(67 136 77 126)(68 135 78 125)(69 134 79 124)(70 133 80 123)
G:=sub<Sym(160)| (1,122)(2,123)(3,124)(4,125)(5,126)(6,127)(7,128)(8,129)(9,130)(10,131)(11,132)(12,133)(13,134)(14,135)(15,136)(16,137)(17,138)(18,139)(19,140)(20,121)(21,99)(22,100)(23,81)(24,82)(25,83)(26,84)(27,85)(28,86)(29,87)(30,88)(31,89)(32,90)(33,91)(34,92)(35,93)(36,94)(37,95)(38,96)(39,97)(40,98)(41,150)(42,151)(43,152)(44,153)(45,154)(46,155)(47,156)(48,157)(49,158)(50,159)(51,160)(52,141)(53,142)(54,143)(55,144)(56,145)(57,146)(58,147)(59,148)(60,149)(61,112)(62,113)(63,114)(64,115)(65,116)(66,117)(67,118)(68,119)(69,120)(70,101)(71,102)(72,103)(73,104)(74,105)(75,106)(76,107)(77,108)(78,109)(79,110)(80,111), (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), (1,159,127,45,11,149,137,55)(2,158,128,44,12,148,138,54)(3,157,129,43,13,147,139,53)(4,156,130,42,14,146,140,52)(5,155,131,41,15,145,121,51)(6,154,132,60,16,144,122,50)(7,153,133,59,17,143,123,49)(8,152,134,58,18,142,124,48)(9,151,135,57,19,141,125,47)(10,150,136,56,20,160,126,46)(21,105,94,79,31,115,84,69)(22,104,95,78,32,114,85,68)(23,103,96,77,33,113,86,67)(24,102,97,76,34,112,87,66)(25,101,98,75,35,111,88,65)(26,120,99,74,36,110,89,64)(27,119,100,73,37,109,90,63)(28,118,81,72,38,108,91,62)(29,117,82,71,39,107,92,61)(30,116,83,70,40,106,93,80), (1,102,11,112)(2,101,12,111)(3,120,13,110)(4,119,14,109)(5,118,15,108)(6,117,16,107)(7,116,17,106)(8,115,18,105)(9,114,19,104)(10,113,20,103)(21,157,31,147)(22,156,32,146)(23,155,33,145)(24,154,34,144)(25,153,35,143)(26,152,36,142)(27,151,37,141)(28,150,38,160)(29,149,39,159)(30,148,40,158)(41,96,51,86)(42,95,52,85)(43,94,53,84)(44,93,54,83)(45,92,55,82)(46,91,56,81)(47,90,57,100)(48,89,58,99)(49,88,59,98)(50,87,60,97)(61,122,71,132)(62,121,72,131)(63,140,73,130)(64,139,74,129)(65,138,75,128)(66,137,76,127)(67,136,77,126)(68,135,78,125)(69,134,79,124)(70,133,80,123)>;
G:=Group( (1,122)(2,123)(3,124)(4,125)(5,126)(6,127)(7,128)(8,129)(9,130)(10,131)(11,132)(12,133)(13,134)(14,135)(15,136)(16,137)(17,138)(18,139)(19,140)(20,121)(21,99)(22,100)(23,81)(24,82)(25,83)(26,84)(27,85)(28,86)(29,87)(30,88)(31,89)(32,90)(33,91)(34,92)(35,93)(36,94)(37,95)(38,96)(39,97)(40,98)(41,150)(42,151)(43,152)(44,153)(45,154)(46,155)(47,156)(48,157)(49,158)(50,159)(51,160)(52,141)(53,142)(54,143)(55,144)(56,145)(57,146)(58,147)(59,148)(60,149)(61,112)(62,113)(63,114)(64,115)(65,116)(66,117)(67,118)(68,119)(69,120)(70,101)(71,102)(72,103)(73,104)(74,105)(75,106)(76,107)(77,108)(78,109)(79,110)(80,111), (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), (1,159,127,45,11,149,137,55)(2,158,128,44,12,148,138,54)(3,157,129,43,13,147,139,53)(4,156,130,42,14,146,140,52)(5,155,131,41,15,145,121,51)(6,154,132,60,16,144,122,50)(7,153,133,59,17,143,123,49)(8,152,134,58,18,142,124,48)(9,151,135,57,19,141,125,47)(10,150,136,56,20,160,126,46)(21,105,94,79,31,115,84,69)(22,104,95,78,32,114,85,68)(23,103,96,77,33,113,86,67)(24,102,97,76,34,112,87,66)(25,101,98,75,35,111,88,65)(26,120,99,74,36,110,89,64)(27,119,100,73,37,109,90,63)(28,118,81,72,38,108,91,62)(29,117,82,71,39,107,92,61)(30,116,83,70,40,106,93,80), (1,102,11,112)(2,101,12,111)(3,120,13,110)(4,119,14,109)(5,118,15,108)(6,117,16,107)(7,116,17,106)(8,115,18,105)(9,114,19,104)(10,113,20,103)(21,157,31,147)(22,156,32,146)(23,155,33,145)(24,154,34,144)(25,153,35,143)(26,152,36,142)(27,151,37,141)(28,150,38,160)(29,149,39,159)(30,148,40,158)(41,96,51,86)(42,95,52,85)(43,94,53,84)(44,93,54,83)(45,92,55,82)(46,91,56,81)(47,90,57,100)(48,89,58,99)(49,88,59,98)(50,87,60,97)(61,122,71,132)(62,121,72,131)(63,140,73,130)(64,139,74,129)(65,138,75,128)(66,137,76,127)(67,136,77,126)(68,135,78,125)(69,134,79,124)(70,133,80,123) );
G=PermutationGroup([[(1,122),(2,123),(3,124),(4,125),(5,126),(6,127),(7,128),(8,129),(9,130),(10,131),(11,132),(12,133),(13,134),(14,135),(15,136),(16,137),(17,138),(18,139),(19,140),(20,121),(21,99),(22,100),(23,81),(24,82),(25,83),(26,84),(27,85),(28,86),(29,87),(30,88),(31,89),(32,90),(33,91),(34,92),(35,93),(36,94),(37,95),(38,96),(39,97),(40,98),(41,150),(42,151),(43,152),(44,153),(45,154),(46,155),(47,156),(48,157),(49,158),(50,159),(51,160),(52,141),(53,142),(54,143),(55,144),(56,145),(57,146),(58,147),(59,148),(60,149),(61,112),(62,113),(63,114),(64,115),(65,116),(66,117),(67,118),(68,119),(69,120),(70,101),(71,102),(72,103),(73,104),(74,105),(75,106),(76,107),(77,108),(78,109),(79,110),(80,111)], [(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)], [(1,159,127,45,11,149,137,55),(2,158,128,44,12,148,138,54),(3,157,129,43,13,147,139,53),(4,156,130,42,14,146,140,52),(5,155,131,41,15,145,121,51),(6,154,132,60,16,144,122,50),(7,153,133,59,17,143,123,49),(8,152,134,58,18,142,124,48),(9,151,135,57,19,141,125,47),(10,150,136,56,20,160,126,46),(21,105,94,79,31,115,84,69),(22,104,95,78,32,114,85,68),(23,103,96,77,33,113,86,67),(24,102,97,76,34,112,87,66),(25,101,98,75,35,111,88,65),(26,120,99,74,36,110,89,64),(27,119,100,73,37,109,90,63),(28,118,81,72,38,108,91,62),(29,117,82,71,39,107,92,61),(30,116,83,70,40,106,93,80)], [(1,102,11,112),(2,101,12,111),(3,120,13,110),(4,119,14,109),(5,118,15,108),(6,117,16,107),(7,116,17,106),(8,115,18,105),(9,114,19,104),(10,113,20,103),(21,157,31,147),(22,156,32,146),(23,155,33,145),(24,154,34,144),(25,153,35,143),(26,152,36,142),(27,151,37,141),(28,150,38,160),(29,149,39,159),(30,148,40,158),(41,96,51,86),(42,95,52,85),(43,94,53,84),(44,93,54,83),(45,92,55,82),(46,91,56,81),(47,90,57,100),(48,89,58,99),(49,88,59,98),(50,87,60,97),(61,122,71,132),(62,121,72,131),(63,140,73,130),(64,139,74,129),(65,138,75,128),(66,137,76,127),(67,136,77,126),(68,135,78,125),(69,134,79,124),(70,133,80,123)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D5 | D10 | D10 | C4×D5 | D20 | C5⋊D4 | C4×D5 | C4.10D4 | C4.12D20 |
| kernel | C2×C4.12D20 | C4.12D20 | C2×C4.Dic5 | C10×M4(2) | C22×Dic10 | C2×Dic10 | C22×Dic5 | C2×C20 | C2×M4(2) | M4(2) | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C10 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 4 | 4 | 2 | 4 | 2 | 4 | 8 | 8 | 4 | 2 | 8 |
Matrix representation of C2×C4.12D20 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 40 | 0 | 0 | 0 | 0 |
| 36 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 11 | 0 | 0 |
| 0 | 0 | 30 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 39 | 11 |
| 0 | 0 | 0 | 0 | 14 | 25 |
| 35 | 3 | 0 | 0 | 0 | 0 |
| 15 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 24 | 4 | 39 |
| 0 | 0 | 17 | 40 | 27 | 39 |
| 0 | 0 | 8 | 17 | 23 | 1 |
| 0 | 0 | 10 | 9 | 3 | 18 |
| 28 | 27 | 0 | 0 | 0 | 0 |
| 12 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 4 | 9 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,36,0,0,0,0,40,6,0,0,0,0,0,0,9,30,0,0,0,0,11,14,0,0,0,0,0,0,39,14,0,0,0,0,11,25],[35,15,0,0,0,0,3,6,0,0,0,0,0,0,1,17,8,10,0,0,24,40,17,9,0,0,4,27,23,3,0,0,39,39,1,18],[28,12,0,0,0,0,27,13,0,0,0,0,0,0,0,32,0,0,0,0,32,0,0,0,0,0,0,0,32,4,0,0,0,0,0,9] >;
C2×C4.12D20 in GAP, Magma, Sage, TeX
C_2\times C_4._{12}D_{20} % in TeX
G:=Group("C2xC4.12D20"); // GroupNames label
G:=SmallGroup(320,763);
// by ID
G=gap.SmallGroup(320,763);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,422,58,1123,136,438,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=1,c^4=d^2=b^10,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^5*c^3>;
// generators/relations