metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊2D20, C40⋊7D4, C4.Q8⋊3D5, C5⋊2(C8⋊2D4), (C2×D40)⋊24C2, C4⋊D20⋊6C2, C4⋊C4.39D10, C4.51(C2×D20), (C2×C8).61D10, C20.131(C2×D4), D20⋊6C4⋊16C2, C20.30(C4○D4), C4.4(Q8⋊2D5), (C2×Dic5).50D4, (C22×D5).32D4, C22.217(D4×D5), C2.22(D40⋊C2), C10.44(C4⋊D4), C2.17(C4⋊D20), C10.70(C8⋊C22), (C2×C20).281C23, (C2×C40).110C22, (C2×D20).79C22, (C5×C4.Q8)⋊3C2, (C2×C8⋊D5)⋊2C2, (C2×C4×D5).37C22, (C2×C10).286(C2×D4), (C5×C4⋊C4).74C22, (C2×C5⋊2C8).58C22, (C2×C4).384(C22×D5), SmallGroup(320,492)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8⋊2D20
G = < a,b,c | a8=b20=c2=1, bab-1=a3, cac=a-1, cbc=b-1 >
Subgroups: 742 in 130 conjugacy classes, 41 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, D4⋊C4, C4.Q8, C4⋊D4, C2×M4(2), C2×D8, C5⋊2C8, C40, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C8⋊2D4, C8⋊D5, D40, C2×C5⋊2C8, D10⋊C4, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×D20, C2×D20, D20⋊6C4, C5×C4.Q8, C4⋊D20, C2×C8⋊D5, C2×D40, C8⋊2D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C8⋊C22, D20, C22×D5, C8⋊2D4, C2×D20, D4×D5, Q8⋊2D5, C4⋊D20, D40⋊C2, C8⋊2D20
►On 160 points
Generators in S160
(1 33 134 102 153 89 54 61)(2 103 55 34 154 62 135 90)(3 35 136 104 155 91 56 63)(4 105 57 36 156 64 137 92)(5 37 138 106 157 93 58 65)(6 107 59 38 158 66 139 94)(7 39 140 108 159 95 60 67)(8 109 41 40 160 68 121 96)(9 21 122 110 141 97 42 69)(10 111 43 22 142 70 123 98)(11 23 124 112 143 99 44 71)(12 113 45 24 144 72 125 100)(13 25 126 114 145 81 46 73)(14 115 47 26 146 74 127 82)(15 27 128 116 147 83 48 75)(16 117 49 28 148 76 129 84)(17 29 130 118 149 85 50 77)(18 119 51 30 150 78 131 86)(19 31 132 120 151 87 52 79)(20 101 53 32 152 80 133 88)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 67)(22 66)(23 65)(24 64)(25 63)(26 62)(27 61)(28 80)(29 79)(30 78)(31 77)(32 76)(33 75)(34 74)(35 73)(36 72)(37 71)(38 70)(39 69)(40 68)(41 121)(42 140)(43 139)(44 138)(45 137)(46 136)(47 135)(48 134)(49 133)(50 132)(51 131)(52 130)(53 129)(54 128)(55 127)(56 126)(57 125)(58 124)(59 123)(60 122)(81 104)(82 103)(83 102)(84 101)(85 120)(86 119)(87 118)(88 117)(89 116)(90 115)(91 114)(92 113)(93 112)(94 111)(95 110)(96 109)(97 108)(98 107)(99 106)(100 105)(141 159)(142 158)(143 157)(144 156)(145 155)(146 154)(147 153)(148 152)(149 151)
G:=sub<Sym(160)| (1,33,134,102,153,89,54,61)(2,103,55,34,154,62,135,90)(3,35,136,104,155,91,56,63)(4,105,57,36,156,64,137,92)(5,37,138,106,157,93,58,65)(6,107,59,38,158,66,139,94)(7,39,140,108,159,95,60,67)(8,109,41,40,160,68,121,96)(9,21,122,110,141,97,42,69)(10,111,43,22,142,70,123,98)(11,23,124,112,143,99,44,71)(12,113,45,24,144,72,125,100)(13,25,126,114,145,81,46,73)(14,115,47,26,146,74,127,82)(15,27,128,116,147,83,48,75)(16,117,49,28,148,76,129,84)(17,29,130,118,149,85,50,77)(18,119,51,30,150,78,131,86)(19,31,132,120,151,87,52,79)(20,101,53,32,152,80,133,88), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,67)(22,66)(23,65)(24,64)(25,63)(26,62)(27,61)(28,80)(29,79)(30,78)(31,77)(32,76)(33,75)(34,74)(35,73)(36,72)(37,71)(38,70)(39,69)(40,68)(41,121)(42,140)(43,139)(44,138)(45,137)(46,136)(47,135)(48,134)(49,133)(50,132)(51,131)(52,130)(53,129)(54,128)(55,127)(56,126)(57,125)(58,124)(59,123)(60,122)(81,104)(82,103)(83,102)(84,101)(85,120)(86,119)(87,118)(88,117)(89,116)(90,115)(91,114)(92,113)(93,112)(94,111)(95,110)(96,109)(97,108)(98,107)(99,106)(100,105)(141,159)(142,158)(143,157)(144,156)(145,155)(146,154)(147,153)(148,152)(149,151)>;
G:=Group( (1,33,134,102,153,89,54,61)(2,103,55,34,154,62,135,90)(3,35,136,104,155,91,56,63)(4,105,57,36,156,64,137,92)(5,37,138,106,157,93,58,65)(6,107,59,38,158,66,139,94)(7,39,140,108,159,95,60,67)(8,109,41,40,160,68,121,96)(9,21,122,110,141,97,42,69)(10,111,43,22,142,70,123,98)(11,23,124,112,143,99,44,71)(12,113,45,24,144,72,125,100)(13,25,126,114,145,81,46,73)(14,115,47,26,146,74,127,82)(15,27,128,116,147,83,48,75)(16,117,49,28,148,76,129,84)(17,29,130,118,149,85,50,77)(18,119,51,30,150,78,131,86)(19,31,132,120,151,87,52,79)(20,101,53,32,152,80,133,88), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,67)(22,66)(23,65)(24,64)(25,63)(26,62)(27,61)(28,80)(29,79)(30,78)(31,77)(32,76)(33,75)(34,74)(35,73)(36,72)(37,71)(38,70)(39,69)(40,68)(41,121)(42,140)(43,139)(44,138)(45,137)(46,136)(47,135)(48,134)(49,133)(50,132)(51,131)(52,130)(53,129)(54,128)(55,127)(56,126)(57,125)(58,124)(59,123)(60,122)(81,104)(82,103)(83,102)(84,101)(85,120)(86,119)(87,118)(88,117)(89,116)(90,115)(91,114)(92,113)(93,112)(94,111)(95,110)(96,109)(97,108)(98,107)(99,106)(100,105)(141,159)(142,158)(143,157)(144,156)(145,155)(146,154)(147,153)(148,152)(149,151) );
G=PermutationGroup([[(1,33,134,102,153,89,54,61),(2,103,55,34,154,62,135,90),(3,35,136,104,155,91,56,63),(4,105,57,36,156,64,137,92),(5,37,138,106,157,93,58,65),(6,107,59,38,158,66,139,94),(7,39,140,108,159,95,60,67),(8,109,41,40,160,68,121,96),(9,21,122,110,141,97,42,69),(10,111,43,22,142,70,123,98),(11,23,124,112,143,99,44,71),(12,113,45,24,144,72,125,100),(13,25,126,114,145,81,46,73),(14,115,47,26,146,74,127,82),(15,27,128,116,147,83,48,75),(16,117,49,28,148,76,129,84),(17,29,130,118,149,85,50,77),(18,119,51,30,150,78,131,86),(19,31,132,120,151,87,52,79),(20,101,53,32,152,80,133,88)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,67),(22,66),(23,65),(24,64),(25,63),(26,62),(27,61),(28,80),(29,79),(30,78),(31,77),(32,76),(33,75),(34,74),(35,73),(36,72),(37,71),(38,70),(39,69),(40,68),(41,121),(42,140),(43,139),(44,138),(45,137),(46,136),(47,135),(48,134),(49,133),(50,132),(51,131),(52,130),(53,129),(54,128),(55,127),(56,126),(57,125),(58,124),(59,123),(60,122),(81,104),(82,103),(83,102),(84,101),(85,120),(86,119),(87,118),(88,117),(89,116),(90,115),(91,114),(92,113),(93,112),(94,111),(95,110),(96,109),(97,108),(98,107),(99,106),(100,105),(141,159),(142,158),(143,157),(144,156),(145,155),(146,154),(147,153),(148,152),(149,151)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 20 | 40 | 40 | 2 | 2 | 8 | 8 | 20 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | C4○D4 | D10 | D10 | D20 | C8⋊C22 | Q8⋊2D5 | D4×D5 | D40⋊C2 |
| kernel | C8⋊2D20 | D20⋊6C4 | C5×C4.Q8 | C4⋊D20 | C2×C8⋊D5 | C2×D40 | C40 | C2×Dic5 | C22×D5 | C4.Q8 | C20 | C4⋊C4 | C2×C8 | C8 | C10 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 8 | 2 | 2 | 2 | 8 |
Matrix representation of C8⋊2D20 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 15 | 32 | 26 |
| 0 | 0 | 26 | 32 | 15 | 9 |
| 0 | 0 | 9 | 15 | 9 | 15 |
| 0 | 0 | 26 | 32 | 26 | 32 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 7 | 1 | 35 |
| 0 | 0 | 34 | 2 | 6 | 0 |
| 0 | 0 | 1 | 35 | 6 | 34 |
| 0 | 0 | 6 | 0 | 7 | 39 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 7 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 34 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,26,9,26,0,0,15,32,15,32,0,0,32,15,9,26,0,0,26,9,15,32],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,35,34,1,6,0,0,7,2,35,0,0,0,1,6,6,7,0,0,35,0,34,39],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,7,1,0,0,0,0,0,0,1,0,0,0,0,0,34,40] >;
C8⋊2D20 in GAP, Magma, Sage, TeX
C_8\rtimes_2D_{20} % in TeX
G:=Group("C8:2D20"); // GroupNames label
G:=SmallGroup(320,492);
// by ID
G=gap.SmallGroup(320,492);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,120,254,555,58,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^8=b^20=c^2=1,b*a*b^-1=a^3,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations