metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊8M4(2), Dic5⋊9M4(2), C20⋊C8⋊15C2, C4⋊2(C22.F5), C22.7(C4⋊F5), (C22×C4).22F5, C23.45(C2×F5), Dic5⋊C8⋊7C2, C5⋊3(C4⋊M4(2)), (C22×C20).24C4, Dic5.37(C2×D4), (C4×Dic5).38C4, (C2×Dic5).38Q8, Dic5.19(C2×Q8), Dic5.36(C4⋊C4), (C2×Dic5).180D4, C10.28(C2×M4(2)), C22.87(C22×F5), C2.18(D5⋊M4(2)), (C22×Dic5).34C4, (C2×Dic5).349C23, (C4×Dic5).348C22, (C22×Dic5).277C22, C2.22(C2×C4⋊F5), C10.21(C2×C4⋊C4), (C2×C5⋊C8).8C22, (C2×C4).144(C2×F5), (C2×C4×Dic5).47C2, (C2×C10).27(C4⋊C4), (C2×C20).111(C2×C4), C2.7(C2×C22.F5), (C2×C22.F5).5C2, (C2×C10).65(C22×C4), (C22×C10).65(C2×C4), (C2×Dic5).187(C2×C4), SmallGroup(320,1096)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20⋊8M4(2)
G = < a,b,c | a20=b8=c2=1, bab-1=a3, ac=ca, cbc=b5 >
Subgroups: 378 in 126 conjugacy classes, 58 normal (32 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C42, C2×C8, M4(2), C22×C4, C22×C4, Dic5, Dic5, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C4⋊C8, C2×C42, C2×M4(2), C5⋊C8, C2×Dic5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C4⋊M4(2), C4×Dic5, C2×C5⋊C8, C22.F5, C22×Dic5, C22×C20, C20⋊C8, Dic5⋊C8, C2×C4×Dic5, C2×C22.F5, C20⋊8M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, M4(2), C22×C4, C2×D4, C2×Q8, F5, C2×C4⋊C4, C2×M4(2), C2×F5, C4⋊M4(2), C4⋊F5, C22.F5, C22×F5, D5⋊M4(2), C2×C4⋊F5, C2×C22.F5, C20⋊8M4(2)
►On 160 points
Generators in S160
(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)
(1 146 33 68 122 91 59 106)(2 153 22 71 123 98 48 109)(3 160 31 74 124 85 57 112)(4 147 40 77 125 92 46 115)(5 154 29 80 126 99 55 118)(6 141 38 63 127 86 44 101)(7 148 27 66 128 93 53 104)(8 155 36 69 129 100 42 107)(9 142 25 72 130 87 51 110)(10 149 34 75 131 94 60 113)(11 156 23 78 132 81 49 116)(12 143 32 61 133 88 58 119)(13 150 21 64 134 95 47 102)(14 157 30 67 135 82 56 105)(15 144 39 70 136 89 45 108)(16 151 28 73 137 96 54 111)(17 158 37 76 138 83 43 114)(18 145 26 79 139 90 52 117)(19 152 35 62 140 97 41 120)(20 159 24 65 121 84 50 103)
(61 119)(62 120)(63 101)(64 102)(65 103)(66 104)(67 105)(68 106)(69 107)(70 108)(71 109)(72 110)(73 111)(74 112)(75 113)(76 114)(77 115)(78 116)(79 117)(80 118)(81 156)(82 157)(83 158)(84 159)(85 160)(86 141)(87 142)(88 143)(89 144)(90 145)(91 146)(92 147)(93 148)(94 149)(95 150)(96 151)(97 152)(98 153)(99 154)(100 155)
G:=sub<Sym(160)| (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), (1,146,33,68,122,91,59,106)(2,153,22,71,123,98,48,109)(3,160,31,74,124,85,57,112)(4,147,40,77,125,92,46,115)(5,154,29,80,126,99,55,118)(6,141,38,63,127,86,44,101)(7,148,27,66,128,93,53,104)(8,155,36,69,129,100,42,107)(9,142,25,72,130,87,51,110)(10,149,34,75,131,94,60,113)(11,156,23,78,132,81,49,116)(12,143,32,61,133,88,58,119)(13,150,21,64,134,95,47,102)(14,157,30,67,135,82,56,105)(15,144,39,70,136,89,45,108)(16,151,28,73,137,96,54,111)(17,158,37,76,138,83,43,114)(18,145,26,79,139,90,52,117)(19,152,35,62,140,97,41,120)(20,159,24,65,121,84,50,103), (61,119)(62,120)(63,101)(64,102)(65,103)(66,104)(67,105)(68,106)(69,107)(70,108)(71,109)(72,110)(73,111)(74,112)(75,113)(76,114)(77,115)(78,116)(79,117)(80,118)(81,156)(82,157)(83,158)(84,159)(85,160)(86,141)(87,142)(88,143)(89,144)(90,145)(91,146)(92,147)(93,148)(94,149)(95,150)(96,151)(97,152)(98,153)(99,154)(100,155)>;
G:=Group( (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), (1,146,33,68,122,91,59,106)(2,153,22,71,123,98,48,109)(3,160,31,74,124,85,57,112)(4,147,40,77,125,92,46,115)(5,154,29,80,126,99,55,118)(6,141,38,63,127,86,44,101)(7,148,27,66,128,93,53,104)(8,155,36,69,129,100,42,107)(9,142,25,72,130,87,51,110)(10,149,34,75,131,94,60,113)(11,156,23,78,132,81,49,116)(12,143,32,61,133,88,58,119)(13,150,21,64,134,95,47,102)(14,157,30,67,135,82,56,105)(15,144,39,70,136,89,45,108)(16,151,28,73,137,96,54,111)(17,158,37,76,138,83,43,114)(18,145,26,79,139,90,52,117)(19,152,35,62,140,97,41,120)(20,159,24,65,121,84,50,103), (61,119)(62,120)(63,101)(64,102)(65,103)(66,104)(67,105)(68,106)(69,107)(70,108)(71,109)(72,110)(73,111)(74,112)(75,113)(76,114)(77,115)(78,116)(79,117)(80,118)(81,156)(82,157)(83,158)(84,159)(85,160)(86,141)(87,142)(88,143)(89,144)(90,145)(91,146)(92,147)(93,148)(94,149)(95,150)(96,151)(97,152)(98,153)(99,154)(100,155) );
G=PermutationGroup([[(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)], [(1,146,33,68,122,91,59,106),(2,153,22,71,123,98,48,109),(3,160,31,74,124,85,57,112),(4,147,40,77,125,92,46,115),(5,154,29,80,126,99,55,118),(6,141,38,63,127,86,44,101),(7,148,27,66,128,93,53,104),(8,155,36,69,129,100,42,107),(9,142,25,72,130,87,51,110),(10,149,34,75,131,94,60,113),(11,156,23,78,132,81,49,116),(12,143,32,61,133,88,58,119),(13,150,21,64,134,95,47,102),(14,157,30,67,135,82,56,105),(15,144,39,70,136,89,45,108),(16,151,28,73,137,96,54,111),(17,158,37,76,138,83,43,114),(18,145,26,79,139,90,52,117),(19,152,35,62,140,97,41,120),(20,159,24,65,121,84,50,103)], [(61,119),(62,120),(63,101),(64,102),(65,103),(66,104),(67,105),(68,106),(69,107),(70,108),(71,109),(72,110),(73,111),(74,112),(75,113),(76,114),(77,115),(78,116),(79,117),(80,118),(81,156),(82,157),(83,158),(84,159),(85,160),(86,141),(87,142),(88,143),(89,144),(90,145),(91,146),(92,147),(93,148),(94,149),(95,150),(96,151),(97,152),(98,153),(99,154),(100,155)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 5 | 8A | ··· | 8H | 10A | ··· | 10G | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 4 | 20 | ··· | 20 | 4 | ··· | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | Q8 | M4(2) | M4(2) | F5 | C2×F5 | C2×F5 | C22.F5 | C4⋊F5 | D5⋊M4(2) |
| kernel | C20⋊8M4(2) | C20⋊C8 | Dic5⋊C8 | C2×C4×Dic5 | C2×C22.F5 | C4×Dic5 | C22×Dic5 | C22×C20 | C2×Dic5 | C2×Dic5 | Dic5 | C20 | C22×C4 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 2 | 1 | 4 | 4 | 4 |
Matrix representation of C20⋊8M4(2) ►in GL6(𝔽41)
| 7 | 2 | 0 | 0 | 0 | 0 |
| 16 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 9 | 9 | 3 |
| 0 | 0 | 32 | 0 | 39 | 4 |
| 0 | 0 | 0 | 0 | 28 | 28 |
| 0 | 0 | 0 | 0 | 13 | 32 |
| 0 | 35 | 0 | 0 | 0 | 0 |
| 7 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 38 | 14 | 31 | 17 |
| 0 | 0 | 37 | 33 | 28 | 0 |
| 0 | 0 | 13 | 8 | 0 | 28 |
| 0 | 0 | 4 | 28 | 3 | 11 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 29 | 20 |
| 0 | 0 | 0 | 1 | 40 | 36 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [7,16,0,0,0,0,2,34,0,0,0,0,0,0,28,32,0,0,0,0,9,0,0,0,0,0,9,39,28,13,0,0,3,4,28,32],[0,7,0,0,0,0,35,0,0,0,0,0,0,0,38,37,13,4,0,0,14,33,8,28,0,0,31,28,0,3,0,0,17,0,28,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,29,40,40,0,0,0,20,36,0,40] >;
C20⋊8M4(2) in GAP, Magma, Sage, TeX
C_{20}\rtimes_8M_4(2) % in TeX
G:=Group("C20:8M4(2)"); // GroupNames label
G:=SmallGroup(320,1096);
// by ID
G=gap.SmallGroup(320,1096);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,253,120,758,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^3,a*c=c*a,c*b*c=b^5>;
// generators/relations