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