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), Dic5.11(C2×Q8), (C2×Dic5).13Q8, 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, (C2×C10).34(C22×C4), (C22×C10).17(C2×C4), (C2×Dic5).51(C2×C4), SmallGroup(320,1034)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 346 in 114 conjugacy classes, 52 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×8], C22, C22 [×2], C22 [×2], C5, C8 [×4], C2×C4 [×2], C2×C4 [×8], C23, C10, C10 [×2], C10 [×2], C42 [×2], C22⋊C4, C22⋊C4, C4⋊C4 [×2], C2×C8 [×6], M4(2) [×2], C22×C4, Dic5 [×2], Dic5 [×2], Dic5 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C4⋊C8 [×4], C42⋊C2, C22×C8, C2×M4(2), C5⋊C8 [×4], C2×Dic5 [×2], C2×Dic5 [×6], C2×C20 [×2], C22×C10, C42.6C22, C4×Dic5 [×2], C10.D4 [×2], C23.D5, C5×C22⋊C4, C2×C5⋊C8 [×2], C2×C5⋊C8 [×2], C2×C5⋊C8 [×2], C22.F5 [×2], C22×Dic5, C20⋊C8 [×2], Dic5⋊C8 [×2], C23.11D10, C22×C5⋊C8, C2×C22.F5, C20⋊C8⋊C2
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, F5, C2×C4⋊C4, C8○D4 [×2], C2×F5 [×3], C42.6C22, C4⋊F5 [×2], C22×F5, C2×C4⋊F5, D4.F5 [×2], C20⋊C8⋊C2
Generators and relations
G = < a,b,c | a20=b8=c2=1, bab-1=a3, cac=ab4, cbc=b5 >
►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)
(1 101 127 95 23 69 152 53)(2 108 136 98 24 76 141 56)(3 115 125 81 25 63 150 59)(4 102 134 84 26 70 159 42)(5 109 123 87 27 77 148 45)(6 116 132 90 28 64 157 48)(7 103 121 93 29 71 146 51)(8 110 130 96 30 78 155 54)(9 117 139 99 31 65 144 57)(10 104 128 82 32 72 153 60)(11 111 137 85 33 79 142 43)(12 118 126 88 34 66 151 46)(13 105 135 91 35 73 160 49)(14 112 124 94 36 80 149 52)(15 119 133 97 37 67 158 55)(16 106 122 100 38 74 147 58)(17 113 131 83 39 61 156 41)(18 120 140 86 40 68 145 44)(19 107 129 89 21 75 154 47)(20 114 138 92 22 62 143 50)
(2 24)(4 26)(6 28)(8 30)(10 32)(12 34)(14 36)(16 38)(18 40)(20 22)(41 83)(43 85)(45 87)(47 89)(49 91)(51 93)(53 95)(55 97)(57 99)(59 81)(61 113)(63 115)(65 117)(67 119)(69 101)(71 103)(73 105)(75 107)(77 109)(79 111)(122 147)(124 149)(126 151)(128 153)(130 155)(132 157)(134 159)(136 141)(138 143)(140 145)
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), (1,101,127,95,23,69,152,53)(2,108,136,98,24,76,141,56)(3,115,125,81,25,63,150,59)(4,102,134,84,26,70,159,42)(5,109,123,87,27,77,148,45)(6,116,132,90,28,64,157,48)(7,103,121,93,29,71,146,51)(8,110,130,96,30,78,155,54)(9,117,139,99,31,65,144,57)(10,104,128,82,32,72,153,60)(11,111,137,85,33,79,142,43)(12,118,126,88,34,66,151,46)(13,105,135,91,35,73,160,49)(14,112,124,94,36,80,149,52)(15,119,133,97,37,67,158,55)(16,106,122,100,38,74,147,58)(17,113,131,83,39,61,156,41)(18,120,140,86,40,68,145,44)(19,107,129,89,21,75,154,47)(20,114,138,92,22,62,143,50), (2,24)(4,26)(6,28)(8,30)(10,32)(12,34)(14,36)(16,38)(18,40)(20,22)(41,83)(43,85)(45,87)(47,89)(49,91)(51,93)(53,95)(55,97)(57,99)(59,81)(61,113)(63,115)(65,117)(67,119)(69,101)(71,103)(73,105)(75,107)(77,109)(79,111)(122,147)(124,149)(126,151)(128,153)(130,155)(132,157)(134,159)(136,141)(138,143)(140,145)>;
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), (1,101,127,95,23,69,152,53)(2,108,136,98,24,76,141,56)(3,115,125,81,25,63,150,59)(4,102,134,84,26,70,159,42)(5,109,123,87,27,77,148,45)(6,116,132,90,28,64,157,48)(7,103,121,93,29,71,146,51)(8,110,130,96,30,78,155,54)(9,117,139,99,31,65,144,57)(10,104,128,82,32,72,153,60)(11,111,137,85,33,79,142,43)(12,118,126,88,34,66,151,46)(13,105,135,91,35,73,160,49)(14,112,124,94,36,80,149,52)(15,119,133,97,37,67,158,55)(16,106,122,100,38,74,147,58)(17,113,131,83,39,61,156,41)(18,120,140,86,40,68,145,44)(19,107,129,89,21,75,154,47)(20,114,138,92,22,62,143,50), (2,24)(4,26)(6,28)(8,30)(10,32)(12,34)(14,36)(16,38)(18,40)(20,22)(41,83)(43,85)(45,87)(47,89)(49,91)(51,93)(53,95)(55,97)(57,99)(59,81)(61,113)(63,115)(65,117)(67,119)(69,101)(71,103)(73,105)(75,107)(77,109)(79,111)(122,147)(124,149)(126,151)(128,153)(130,155)(132,157)(134,159)(136,141)(138,143)(140,145) );
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)], [(1,101,127,95,23,69,152,53),(2,108,136,98,24,76,141,56),(3,115,125,81,25,63,150,59),(4,102,134,84,26,70,159,42),(5,109,123,87,27,77,148,45),(6,116,132,90,28,64,157,48),(7,103,121,93,29,71,146,51),(8,110,130,96,30,78,155,54),(9,117,139,99,31,65,144,57),(10,104,128,82,32,72,153,60),(11,111,137,85,33,79,142,43),(12,118,126,88,34,66,151,46),(13,105,135,91,35,73,160,49),(14,112,124,94,36,80,149,52),(15,119,133,97,37,67,158,55),(16,106,122,100,38,74,147,58),(17,113,131,83,39,61,156,41),(18,120,140,86,40,68,145,44),(19,107,129,89,21,75,154,47),(20,114,138,92,22,62,143,50)], [(2,24),(4,26),(6,28),(8,30),(10,32),(12,34),(14,36),(16,38),(18,40),(20,22),(41,83),(43,85),(45,87),(47,89),(49,91),(51,93),(53,95),(55,97),(57,99),(59,81),(61,113),(63,115),(65,117),(67,119),(69,101),(71,103),(73,105),(75,107),(77,109),(79,111),(122,147),(124,149),(126,151),(128,153),(130,155),(132,157),(134,159),(136,141),(138,143),(140,145)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
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ℤ
𝔽
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