metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C20).33D4, (C2×C4).22D20, (C22×D5).2Q8, C22.44(Q8×D5), C2.4(C20⋊4D4), 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×C20).33D4
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, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, Dic5, C20, D10, C2×C10, C2×C10, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C23.4Q8, C4⋊Dic5, D10⋊C4, C22×Dic5, C22×C20, C23×D5, C5×C2.C42, C2×C4⋊Dic5, C2×D10⋊C4, (C2×C20).33D4
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C22.D4, C4⋊1D4, D20, C22×D5, C23.4Q8, C2×D20, D4⋊2D5, Q8×D5, C20⋊4D4, C22.D20, D10⋊2Q8, (C2×C20).33D4
►On 160 points
Generators in S160
(1 120)(2 101)(3 102)(4 103)(5 104)(6 105)(7 106)(8 107)(9 108)(10 109)(11 110)(12 111)(13 112)(14 113)(15 114)(16 115)(17 116)(18 117)(19 118)(20 119)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 71)(28 72)(29 73)(30 74)(31 75)(32 76)(33 77)(34 78)(35 79)(36 80)(37 61)(38 62)(39 63)(40 64)(41 126)(42 127)(43 128)(44 129)(45 130)(46 131)(47 132)(48 133)(49 134)(50 135)(51 136)(52 137)(53 138)(54 139)(55 140)(56 121)(57 122)(58 123)(59 124)(60 125)(81 150)(82 151)(83 152)(84 153)(85 154)(86 155)(87 156)(88 157)(89 158)(90 159)(91 160)(92 141)(93 142)(94 143)(95 144)(96 145)(97 146)(98 147)(99 148)(100 149)
(1 96 80 127)(2 146 61 43)(3 98 62 129)(4 148 63 45)(5 100 64 131)(6 150 65 47)(7 82 66 133)(8 152 67 49)(9 84 68 135)(10 154 69 51)(11 86 70 137)(12 156 71 53)(13 88 72 139)(14 158 73 55)(15 90 74 121)(16 160 75 57)(17 92 76 123)(18 142 77 59)(19 94 78 125)(20 144 79 41)(21 132 105 81)(22 48 106 151)(23 134 107 83)(24 50 108 153)(25 136 109 85)(26 52 110 155)(27 138 111 87)(28 54 112 157)(29 140 113 89)(30 56 114 159)(31 122 115 91)(32 58 116 141)(33 124 117 93)(34 60 118 143)(35 126 119 95)(36 42 120 145)(37 128 101 97)(38 44 102 147)(39 130 103 99)(40 46 104 149)
(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)
(1 119 120 20)(2 19 101 118)(3 117 102 18)(4 17 103 116)(5 115 104 16)(6 15 105 114)(7 113 106 14)(8 13 107 112)(9 111 108 12)(10 11 109 110)(21 30 65 74)(22 73 66 29)(23 28 67 72)(24 71 68 27)(25 26 69 70)(31 40 75 64)(32 63 76 39)(33 38 77 62)(34 61 78 37)(35 36 79 80)(41 96 126 145)(42 144 127 95)(43 94 128 143)(44 142 129 93)(45 92 130 141)(46 160 131 91)(47 90 132 159)(48 158 133 89)(49 88 134 157)(50 156 135 87)(51 86 136 155)(52 154 137 85)(53 84 138 153)(54 152 139 83)(55 82 140 151)(56 150 121 81)(57 100 122 149)(58 148 123 99)(59 98 124 147)(60 146 125 97)
G:=sub<Sym(160)| (1,120)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,109)(11,110)(12,111)(13,112)(14,113)(15,114)(16,115)(17,116)(18,117)(19,118)(20,119)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(37,61)(38,62)(39,63)(40,64)(41,126)(42,127)(43,128)(44,129)(45,130)(46,131)(47,132)(48,133)(49,134)(50,135)(51,136)(52,137)(53,138)(54,139)(55,140)(56,121)(57,122)(58,123)(59,124)(60,125)(81,150)(82,151)(83,152)(84,153)(85,154)(86,155)(87,156)(88,157)(89,158)(90,159)(91,160)(92,141)(93,142)(94,143)(95,144)(96,145)(97,146)(98,147)(99,148)(100,149), (1,96,80,127)(2,146,61,43)(3,98,62,129)(4,148,63,45)(5,100,64,131)(6,150,65,47)(7,82,66,133)(8,152,67,49)(9,84,68,135)(10,154,69,51)(11,86,70,137)(12,156,71,53)(13,88,72,139)(14,158,73,55)(15,90,74,121)(16,160,75,57)(17,92,76,123)(18,142,77,59)(19,94,78,125)(20,144,79,41)(21,132,105,81)(22,48,106,151)(23,134,107,83)(24,50,108,153)(25,136,109,85)(26,52,110,155)(27,138,111,87)(28,54,112,157)(29,140,113,89)(30,56,114,159)(31,122,115,91)(32,58,116,141)(33,124,117,93)(34,60,118,143)(35,126,119,95)(36,42,120,145)(37,128,101,97)(38,44,102,147)(39,130,103,99)(40,46,104,149), (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), (1,119,120,20)(2,19,101,118)(3,117,102,18)(4,17,103,116)(5,115,104,16)(6,15,105,114)(7,113,106,14)(8,13,107,112)(9,111,108,12)(10,11,109,110)(21,30,65,74)(22,73,66,29)(23,28,67,72)(24,71,68,27)(25,26,69,70)(31,40,75,64)(32,63,76,39)(33,38,77,62)(34,61,78,37)(35,36,79,80)(41,96,126,145)(42,144,127,95)(43,94,128,143)(44,142,129,93)(45,92,130,141)(46,160,131,91)(47,90,132,159)(48,158,133,89)(49,88,134,157)(50,156,135,87)(51,86,136,155)(52,154,137,85)(53,84,138,153)(54,152,139,83)(55,82,140,151)(56,150,121,81)(57,100,122,149)(58,148,123,99)(59,98,124,147)(60,146,125,97)>;
G:=Group( (1,120)(2,101)(3,102)(4,103)(5,104)(6,105)(7,106)(8,107)(9,108)(10,109)(11,110)(12,111)(13,112)(14,113)(15,114)(16,115)(17,116)(18,117)(19,118)(20,119)(21,65)(22,66)(23,67)(24,68)(25,69)(26,70)(27,71)(28,72)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(37,61)(38,62)(39,63)(40,64)(41,126)(42,127)(43,128)(44,129)(45,130)(46,131)(47,132)(48,133)(49,134)(50,135)(51,136)(52,137)(53,138)(54,139)(55,140)(56,121)(57,122)(58,123)(59,124)(60,125)(81,150)(82,151)(83,152)(84,153)(85,154)(86,155)(87,156)(88,157)(89,158)(90,159)(91,160)(92,141)(93,142)(94,143)(95,144)(96,145)(97,146)(98,147)(99,148)(100,149), (1,96,80,127)(2,146,61,43)(3,98,62,129)(4,148,63,45)(5,100,64,131)(6,150,65,47)(7,82,66,133)(8,152,67,49)(9,84,68,135)(10,154,69,51)(11,86,70,137)(12,156,71,53)(13,88,72,139)(14,158,73,55)(15,90,74,121)(16,160,75,57)(17,92,76,123)(18,142,77,59)(19,94,78,125)(20,144,79,41)(21,132,105,81)(22,48,106,151)(23,134,107,83)(24,50,108,153)(25,136,109,85)(26,52,110,155)(27,138,111,87)(28,54,112,157)(29,140,113,89)(30,56,114,159)(31,122,115,91)(32,58,116,141)(33,124,117,93)(34,60,118,143)(35,126,119,95)(36,42,120,145)(37,128,101,97)(38,44,102,147)(39,130,103,99)(40,46,104,149), (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), (1,119,120,20)(2,19,101,118)(3,117,102,18)(4,17,103,116)(5,115,104,16)(6,15,105,114)(7,113,106,14)(8,13,107,112)(9,111,108,12)(10,11,109,110)(21,30,65,74)(22,73,66,29)(23,28,67,72)(24,71,68,27)(25,26,69,70)(31,40,75,64)(32,63,76,39)(33,38,77,62)(34,61,78,37)(35,36,79,80)(41,96,126,145)(42,144,127,95)(43,94,128,143)(44,142,129,93)(45,92,130,141)(46,160,131,91)(47,90,132,159)(48,158,133,89)(49,88,134,157)(50,156,135,87)(51,86,136,155)(52,154,137,85)(53,84,138,153)(54,152,139,83)(55,82,140,151)(56,150,121,81)(57,100,122,149)(58,148,123,99)(59,98,124,147)(60,146,125,97) );
G=PermutationGroup([[(1,120),(2,101),(3,102),(4,103),(5,104),(6,105),(7,106),(8,107),(9,108),(10,109),(11,110),(12,111),(13,112),(14,113),(15,114),(16,115),(17,116),(18,117),(19,118),(20,119),(21,65),(22,66),(23,67),(24,68),(25,69),(26,70),(27,71),(28,72),(29,73),(30,74),(31,75),(32,76),(33,77),(34,78),(35,79),(36,80),(37,61),(38,62),(39,63),(40,64),(41,126),(42,127),(43,128),(44,129),(45,130),(46,131),(47,132),(48,133),(49,134),(50,135),(51,136),(52,137),(53,138),(54,139),(55,140),(56,121),(57,122),(58,123),(59,124),(60,125),(81,150),(82,151),(83,152),(84,153),(85,154),(86,155),(87,156),(88,157),(89,158),(90,159),(91,160),(92,141),(93,142),(94,143),(95,144),(96,145),(97,146),(98,147),(99,148),(100,149)], [(1,96,80,127),(2,146,61,43),(3,98,62,129),(4,148,63,45),(5,100,64,131),(6,150,65,47),(7,82,66,133),(8,152,67,49),(9,84,68,135),(10,154,69,51),(11,86,70,137),(12,156,71,53),(13,88,72,139),(14,158,73,55),(15,90,74,121),(16,160,75,57),(17,92,76,123),(18,142,77,59),(19,94,78,125),(20,144,79,41),(21,132,105,81),(22,48,106,151),(23,134,107,83),(24,50,108,153),(25,136,109,85),(26,52,110,155),(27,138,111,87),(28,54,112,157),(29,140,113,89),(30,56,114,159),(31,122,115,91),(32,58,116,141),(33,124,117,93),(34,60,118,143),(35,126,119,95),(36,42,120,145),(37,128,101,97),(38,44,102,147),(39,130,103,99),(40,46,104,149)], [(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)], [(1,119,120,20),(2,19,101,118),(3,117,102,18),(4,17,103,116),(5,115,104,16),(6,15,105,114),(7,113,106,14),(8,13,107,112),(9,111,108,12),(10,11,109,110),(21,30,65,74),(22,73,66,29),(23,28,67,72),(24,71,68,27),(25,26,69,70),(31,40,75,64),(32,63,76,39),(33,38,77,62),(34,61,78,37),(35,36,79,80),(41,96,126,145),(42,144,127,95),(43,94,128,143),(44,142,129,93),(45,92,130,141),(46,160,131,91),(47,90,132,159),(48,158,133,89),(49,88,134,157),(50,156,135,87),(51,86,136,155),(52,154,137,85),(53,84,138,153),(54,152,139,83),(55,82,140,151),(56,150,121,81),(57,100,122,149),(58,148,123,99),(59,98,124,147),(60,146,125,97)]])
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×C20).33D4 | 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×C20).33D4 ►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×C20).33D4 in GAP, Magma, Sage, TeX
(C_2\times C_{20})._{33}D_4 % in TeX
G:=Group("(C2xC20).33D4"); // 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