metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊C8⋊11C2, C22⋊C4.5F5, C10.6(C8○D4), C2.9(D4.F5), C23.D5.3C4, C23.26(C2×F5), Dic5⋊C8⋊8C2, Dic5.9(C4⋊C4), (C2×Dic5).13Q8, Dic5.11(C2×Q8), Dic5.29(C2×D4), C10.D4.3C4, C22.12(C4⋊F5), (C2×Dic5).114D4, C22.72(C22×F5), C5⋊1(C42.6C22), (C2×Dic5).326C23, (C4×Dic5).242C22, C23.11D10.6C2, (C22×Dic5).181C22, C10.7(C2×C4⋊C4), C2.10(C2×C4⋊F5), (C2×C5⋊C8).5C22, (C22×C5⋊C8).3C2, (C2×C4).23(C2×F5), (C2×C10).5(C4⋊C4), (C2×C20).81(C2×C4), (C5×C22⋊C4).3C4, (C2×C22.F5).4C2, (C22×C10).17(C2×C4), (C2×C10).34(C22×C4), (C2×Dic5).51(C2×C4), SmallGroup(320,1034)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20⋊C8⋊C2
G = < a,b,c | a20=b8=c2=1, bab-1=a3, cac=ab4, cbc=b5 >
Subgroups: 346 in 114 conjugacy classes, 52 normal (22 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, Dic5, Dic5, Dic5, C20, C2×C10, C2×C10, C2×C10, C4⋊C8, C42⋊C2, C22×C8, C2×M4(2), C5⋊C8, C2×Dic5, C2×Dic5, C2×C20, C22×C10, C42.6C22, C4×Dic5, C10.D4, C23.D5, C5×C22⋊C4, C2×C5⋊C8, C2×C5⋊C8, C2×C5⋊C8, C22.F5, C22×Dic5, C20⋊C8, Dic5⋊C8, C23.11D10, C22×C5⋊C8, C2×C22.F5, C20⋊C8⋊C2
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, F5, C2×C4⋊C4, C8○D4, C2×F5, C42.6C22, C4⋊F5, C22×F5, C2×C4⋊F5, D4.F5, C20⋊C8⋊C2
►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 51 65 125 102 26 100 148)(2 58 74 128 103 33 89 151)(3 45 63 131 104 40 98 154)(4 52 72 134 105 27 87 157)(5 59 61 137 106 34 96 160)(6 46 70 140 107 21 85 143)(7 53 79 123 108 28 94 146)(8 60 68 126 109 35 83 149)(9 47 77 129 110 22 92 152)(10 54 66 132 111 29 81 155)(11 41 75 135 112 36 90 158)(12 48 64 138 113 23 99 141)(13 55 73 121 114 30 88 144)(14 42 62 124 115 37 97 147)(15 49 71 127 116 24 86 150)(16 56 80 130 117 31 95 153)(17 43 69 133 118 38 84 156)(18 50 78 136 119 25 93 159)(19 57 67 139 120 32 82 142)(20 44 76 122 101 39 91 145)
(2 103)(4 105)(6 107)(8 109)(10 111)(12 113)(14 115)(16 117)(18 119)(20 101)(22 47)(24 49)(26 51)(28 53)(30 55)(32 57)(34 59)(36 41)(38 43)(40 45)(62 97)(64 99)(66 81)(68 83)(70 85)(72 87)(74 89)(76 91)(78 93)(80 95)(121 144)(123 146)(125 148)(127 150)(129 152)(131 154)(133 156)(135 158)(137 160)(139 142)
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), (1,51,65,125,102,26,100,148)(2,58,74,128,103,33,89,151)(3,45,63,131,104,40,98,154)(4,52,72,134,105,27,87,157)(5,59,61,137,106,34,96,160)(6,46,70,140,107,21,85,143)(7,53,79,123,108,28,94,146)(8,60,68,126,109,35,83,149)(9,47,77,129,110,22,92,152)(10,54,66,132,111,29,81,155)(11,41,75,135,112,36,90,158)(12,48,64,138,113,23,99,141)(13,55,73,121,114,30,88,144)(14,42,62,124,115,37,97,147)(15,49,71,127,116,24,86,150)(16,56,80,130,117,31,95,153)(17,43,69,133,118,38,84,156)(18,50,78,136,119,25,93,159)(19,57,67,139,120,32,82,142)(20,44,76,122,101,39,91,145), (2,103)(4,105)(6,107)(8,109)(10,111)(12,113)(14,115)(16,117)(18,119)(20,101)(22,47)(24,49)(26,51)(28,53)(30,55)(32,57)(34,59)(36,41)(38,43)(40,45)(62,97)(64,99)(66,81)(68,83)(70,85)(72,87)(74,89)(76,91)(78,93)(80,95)(121,144)(123,146)(125,148)(127,150)(129,152)(131,154)(133,156)(135,158)(137,160)(139,142)>;
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), (1,51,65,125,102,26,100,148)(2,58,74,128,103,33,89,151)(3,45,63,131,104,40,98,154)(4,52,72,134,105,27,87,157)(5,59,61,137,106,34,96,160)(6,46,70,140,107,21,85,143)(7,53,79,123,108,28,94,146)(8,60,68,126,109,35,83,149)(9,47,77,129,110,22,92,152)(10,54,66,132,111,29,81,155)(11,41,75,135,112,36,90,158)(12,48,64,138,113,23,99,141)(13,55,73,121,114,30,88,144)(14,42,62,124,115,37,97,147)(15,49,71,127,116,24,86,150)(16,56,80,130,117,31,95,153)(17,43,69,133,118,38,84,156)(18,50,78,136,119,25,93,159)(19,57,67,139,120,32,82,142)(20,44,76,122,101,39,91,145), (2,103)(4,105)(6,107)(8,109)(10,111)(12,113)(14,115)(16,117)(18,119)(20,101)(22,47)(24,49)(26,51)(28,53)(30,55)(32,57)(34,59)(36,41)(38,43)(40,45)(62,97)(64,99)(66,81)(68,83)(70,85)(72,87)(74,89)(76,91)(78,93)(80,95)(121,144)(123,146)(125,148)(127,150)(129,152)(131,154)(133,156)(135,158)(137,160)(139,142) );
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)], [(1,51,65,125,102,26,100,148),(2,58,74,128,103,33,89,151),(3,45,63,131,104,40,98,154),(4,52,72,134,105,27,87,157),(5,59,61,137,106,34,96,160),(6,46,70,140,107,21,85,143),(7,53,79,123,108,28,94,146),(8,60,68,126,109,35,83,149),(9,47,77,129,110,22,92,152),(10,54,66,132,111,29,81,155),(11,41,75,135,112,36,90,158),(12,48,64,138,113,23,99,141),(13,55,73,121,114,30,88,144),(14,42,62,124,115,37,97,147),(15,49,71,127,116,24,86,150),(16,56,80,130,117,31,95,153),(17,43,69,133,118,38,84,156),(18,50,78,136,119,25,93,159),(19,57,67,139,120,32,82,142),(20,44,76,122,101,39,91,145)], [(2,103),(4,105),(6,107),(8,109),(10,111),(12,113),(14,115),(16,117),(18,119),(20,101),(22,47),(24,49),(26,51),(28,53),(30,55),(32,57),(34,59),(36,41),(38,43),(40,45),(62,97),(64,99),(66,81),(68,83),(70,85),(72,87),(74,89),(76,91),(78,93),(80,95),(121,144),(123,146),(125,148),(127,150),(129,152),(131,154),(133,156),(135,158),(137,160),(139,142)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5 | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | 10B | 10C | 10D | 10E | 20A | 20B | 20C | 20D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 20 | 20 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | Q8 | C8○D4 | F5 | C2×F5 | C2×F5 | C4⋊F5 | D4.F5 |
| kernel | C20⋊C8⋊C2 | C20⋊C8 | Dic5⋊C8 | C23.11D10 | C22×C5⋊C8 | C2×C22.F5 | C10.D4 | C23.D5 | C5×C22⋊C4 | C2×Dic5 | C2×Dic5 | C10 | C22⋊C4 | C2×C4 | C23 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 8 | 1 | 2 | 1 | 4 | 2 |
Matrix representation of C20⋊C8⋊C2 ►in GL6(𝔽41)
| 0 | 40 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 34 | 14 | 0 |
| 0 | 0 | 34 | 0 | 7 | 14 |
| 0 | 0 | 34 | 27 | 14 | 7 |
| 0 | 0 | 7 | 27 | 0 | 14 |
| 0 | 38 | 0 | 0 | 0 | 0 |
| 38 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 21 | 33 | 34 |
| 0 | 0 | 23 | 14 | 37 | 24 |
| 0 | 0 | 27 | 4 | 17 | 16 |
| 0 | 0 | 7 | 37 | 10 | 20 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 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 | 1 |
G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,40,0,0,0,0,0,0,0,7,34,34,7,0,0,34,0,27,27,0,0,14,7,14,0,0,0,0,14,7,14],[0,38,0,0,0,0,38,0,0,0,0,0,0,0,31,23,27,7,0,0,21,14,4,37,0,0,33,37,17,10,0,0,34,24,16,20],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C20⋊C8⋊C2 in GAP, Magma, Sage, TeX
C_{20}\rtimes C_8\rtimes C_2 % in TeX
G:=Group("C20:C8:C2"); // GroupNames label
G:=SmallGroup(320,1034);
// by ID
G=gap.SmallGroup(320,1034);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,100,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^3,c*a*c=a*b^4,c*b*c=b^5>;
// generators/relations