direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C20.C8, C10⋊2M5(2), C5⋊C16⋊3C22, (C2×C20).5C8, C5⋊3(C2×M5(2)), C20.43(C2×C8), C23.3(C5⋊C8), (C22×C10).6C8, (C22×C4).18F5, C4.55(C22×F5), C20.95(C22×C4), C10.17(C22×C8), (C22×C20).27C4, C5⋊2C8.40C23, C4.9(C2×C5⋊C8), (C2×C5⋊C16)⋊8C2, (C2×C4).6(C5⋊C8), C2.3(C22×C5⋊C8), C22.5(C2×C5⋊C8), (C2×C5⋊2C8).28C4, (C2×C10).32(C2×C8), C5⋊2C8.56(C2×C4), (C2×C4).168(C2×F5), (C2×C20).149(C2×C4), (C22×C5⋊2C8).21C2, (C2×C5⋊2C8).339C22, SmallGroup(320,1081)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 186 in 90 conjugacy classes, 60 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×2], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C23, C10, C10 [×2], C10 [×2], C16 [×4], C2×C8 [×6], C22×C4, C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C16 [×2], M5(2) [×4], C22×C8, C5⋊2C8 [×2], C5⋊2C8 [×2], C2×C20 [×2], C2×C20 [×4], C22×C10, C2×M5(2), C5⋊C16 [×4], C2×C5⋊2C8 [×2], C2×C5⋊2C8 [×4], C22×C20, C2×C5⋊C16 [×2], C20.C8 [×4], C22×C5⋊2C8, C2×C20.C8
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, C2×C8 [×6], C22×C4, F5, M5(2) [×2], C22×C8, C5⋊C8 [×4], C2×F5 [×3], C2×M5(2), C2×C5⋊C8 [×6], C22×F5, C20.C8 [×2], C22×C5⋊C8, C2×C20.C8
Generators and relations
G = < a,b,c | a2=b20=1, c8=b10, ab=ba, ac=ca, cbc-1=b3 >
►On 160 points
Generators in S160
(1 53)(2 54)(3 55)(4 56)(5 57)(6 58)(7 59)(8 60)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 51)(20 52)(21 130)(22 131)(23 132)(24 133)(25 134)(26 135)(27 136)(28 137)(29 138)(30 139)(31 140)(32 121)(33 122)(34 123)(35 124)(36 125)(37 126)(38 127)(39 128)(40 129)(61 142)(62 143)(63 144)(64 145)(65 146)(66 147)(67 148)(68 149)(69 150)(70 151)(71 152)(72 153)(73 154)(74 155)(75 156)(76 157)(77 158)(78 159)(79 160)(80 141)(81 105)(82 106)(83 107)(84 108)(85 109)(86 110)(87 111)(88 112)(89 113)(90 114)(91 115)(92 116)(93 117)(94 118)(95 119)(96 120)(97 101)(98 102)(99 103)(100 104)
(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)
(1 154 139 96 58 68 35 115 11 144 129 86 48 78 25 105)(2 141 128 99 59 75 24 118 12 151 138 89 49 65 34 108)(3 148 137 82 60 62 33 101 13 158 127 92 50 72 23 111)(4 155 126 85 41 69 22 104 14 145 136 95 51 79 32 114)(5 142 135 88 42 76 31 107 15 152 125 98 52 66 21 117)(6 149 124 91 43 63 40 110 16 159 134 81 53 73 30 120)(7 156 133 94 44 70 29 113 17 146 123 84 54 80 39 103)(8 143 122 97 45 77 38 116 18 153 132 87 55 67 28 106)(9 150 131 100 46 64 27 119 19 160 121 90 56 74 37 109)(10 157 140 83 47 71 36 102 20 147 130 93 57 61 26 112)
G:=sub<Sym(160)| (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,49)(18,50)(19,51)(20,52)(21,130)(22,131)(23,132)(24,133)(25,134)(26,135)(27,136)(28,137)(29,138)(30,139)(31,140)(32,121)(33,122)(34,123)(35,124)(36,125)(37,126)(38,127)(39,128)(40,129)(61,142)(62,143)(63,144)(64,145)(65,146)(66,147)(67,148)(68,149)(69,150)(70,151)(71,152)(72,153)(73,154)(74,155)(75,156)(76,157)(77,158)(78,159)(79,160)(80,141)(81,105)(82,106)(83,107)(84,108)(85,109)(86,110)(87,111)(88,112)(89,113)(90,114)(91,115)(92,116)(93,117)(94,118)(95,119)(96,120)(97,101)(98,102)(99,103)(100,104), (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), (1,154,139,96,58,68,35,115,11,144,129,86,48,78,25,105)(2,141,128,99,59,75,24,118,12,151,138,89,49,65,34,108)(3,148,137,82,60,62,33,101,13,158,127,92,50,72,23,111)(4,155,126,85,41,69,22,104,14,145,136,95,51,79,32,114)(5,142,135,88,42,76,31,107,15,152,125,98,52,66,21,117)(6,149,124,91,43,63,40,110,16,159,134,81,53,73,30,120)(7,156,133,94,44,70,29,113,17,146,123,84,54,80,39,103)(8,143,122,97,45,77,38,116,18,153,132,87,55,67,28,106)(9,150,131,100,46,64,27,119,19,160,121,90,56,74,37,109)(10,157,140,83,47,71,36,102,20,147,130,93,57,61,26,112)>;
G:=Group( (1,53)(2,54)(3,55)(4,56)(5,57)(6,58)(7,59)(8,60)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,49)(18,50)(19,51)(20,52)(21,130)(22,131)(23,132)(24,133)(25,134)(26,135)(27,136)(28,137)(29,138)(30,139)(31,140)(32,121)(33,122)(34,123)(35,124)(36,125)(37,126)(38,127)(39,128)(40,129)(61,142)(62,143)(63,144)(64,145)(65,146)(66,147)(67,148)(68,149)(69,150)(70,151)(71,152)(72,153)(73,154)(74,155)(75,156)(76,157)(77,158)(78,159)(79,160)(80,141)(81,105)(82,106)(83,107)(84,108)(85,109)(86,110)(87,111)(88,112)(89,113)(90,114)(91,115)(92,116)(93,117)(94,118)(95,119)(96,120)(97,101)(98,102)(99,103)(100,104), (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), (1,154,139,96,58,68,35,115,11,144,129,86,48,78,25,105)(2,141,128,99,59,75,24,118,12,151,138,89,49,65,34,108)(3,148,137,82,60,62,33,101,13,158,127,92,50,72,23,111)(4,155,126,85,41,69,22,104,14,145,136,95,51,79,32,114)(5,142,135,88,42,76,31,107,15,152,125,98,52,66,21,117)(6,149,124,91,43,63,40,110,16,159,134,81,53,73,30,120)(7,156,133,94,44,70,29,113,17,146,123,84,54,80,39,103)(8,143,122,97,45,77,38,116,18,153,132,87,55,67,28,106)(9,150,131,100,46,64,27,119,19,160,121,90,56,74,37,109)(10,157,140,83,47,71,36,102,20,147,130,93,57,61,26,112) );
G=PermutationGroup([(1,53),(2,54),(3,55),(4,56),(5,57),(6,58),(7,59),(8,60),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(17,49),(18,50),(19,51),(20,52),(21,130),(22,131),(23,132),(24,133),(25,134),(26,135),(27,136),(28,137),(29,138),(30,139),(31,140),(32,121),(33,122),(34,123),(35,124),(36,125),(37,126),(38,127),(39,128),(40,129),(61,142),(62,143),(63,144),(64,145),(65,146),(66,147),(67,148),(68,149),(69,150),(70,151),(71,152),(72,153),(73,154),(74,155),(75,156),(76,157),(77,158),(78,159),(79,160),(80,141),(81,105),(82,106),(83,107),(84,108),(85,109),(86,110),(87,111),(88,112),(89,113),(90,114),(91,115),(92,116),(93,117),(94,118),(95,119),(96,120),(97,101),(98,102),(99,103),(100,104)], [(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)], [(1,154,139,96,58,68,35,115,11,144,129,86,48,78,25,105),(2,141,128,99,59,75,24,118,12,151,138,89,49,65,34,108),(3,148,137,82,60,62,33,101,13,158,127,92,50,72,23,111),(4,155,126,85,41,69,22,104,14,145,136,95,51,79,32,114),(5,142,135,88,42,76,31,107,15,152,125,98,52,66,21,117),(6,149,124,91,43,63,40,110,16,159,134,81,53,73,30,120),(7,156,133,94,44,70,29,113,17,146,123,84,54,80,39,103),(8,143,122,97,45,77,38,116,18,153,132,87,55,67,28,106),(9,150,131,100,46,64,27,119,19,160,121,90,56,74,37,109),(10,157,140,83,47,71,36,102,20,147,130,93,57,61,26,112)])
Matrix representation ►G ⊆ GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 177 | 0 | 0 | 0 | 0 | 0 |
| 236 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 131 | 64 | 0 | 0 |
| 0 | 0 | 177 | 0 | 0 | 0 |
| 0 | 0 | 116 | 117 | 0 | 110 |
| 0 | 0 | 58 | 7 | 195 | 195 |
| 177 | 96 | 0 | 0 | 0 | 0 |
| 43 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 190 | 240 |
| 0 | 0 | 64 | 64 | 2 | 1 |
| 0 | 0 | 83 | 185 | 177 | 0 |
| 0 | 0 | 207 | 215 | 131 | 0 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[177,236,0,0,0,0,0,64,0,0,0,0,0,0,131,177,116,58,0,0,64,0,117,7,0,0,0,0,0,195,0,0,0,0,110,195],[177,43,0,0,0,0,96,64,0,0,0,0,0,0,0,64,83,207,0,0,0,64,185,215,0,0,190,2,177,131,0,0,240,1,0,0] >;
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | ··· | 10G | 16A | ··· | 16P | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 4 | ··· | 4 | 10 | ··· | 10 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | M5(2) | F5 | C5⋊C8 | C2×F5 | C5⋊C8 | C20.C8 |
| kernel | C2×C20.C8 | C2×C5⋊C16 | C20.C8 | C22×C5⋊2C8 | C2×C5⋊2C8 | C22×C20 | C2×C20 | C22×C10 | C10 | C22×C4 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 4 | 1 | 6 | 2 | 12 | 4 | 8 | 1 | 3 | 3 | 1 | 8 |
In GAP, Magma, Sage, TeX
C_2\times C_{20}.C_8 % in TeX
G:=Group("C2xC20.C8"); // GroupNames label
G:=SmallGroup(320,1081);
// by ID
G=gap.SmallGroup(320,1081);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,80,102,6278,1595]);
// Polycyclic
G:=Group<a,b,c|a^2=b^20=1,c^8=b^10,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
×
⋊
ℤ
𝔽
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