metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.34M4(2), (C2×Dic5)⋊9C8, (C22×C4).15F5, C23.41(C2×F5), (C22×C20).35C4, C10.15(C22×C8), C22.6(D5⋊C8), (C4×Dic5).31C4, Dic5.19(C2×C8), Dic5⋊C8⋊16C2, C5⋊4(C42.12C4), C10.26(C2×M4(2)), C4.12(C22.F5), C23.2F5.7C2, Dic5.30(C4○D4), C22.48(C22×F5), (C22×Dic5).32C4, C10.14(C42⋊C2), (C4×Dic5).327C22, (C2×Dic5).345C23, C2.6(D10.C23), (C22×Dic5).273C22, (C4×C5⋊C8)⋊18C2, C2.16(C2×D5⋊C8), (C2×C10).15(C2×C8), (C2×C5⋊C8).38C22, (C2×C4).108(C2×F5), (C2×C4×Dic5).52C2, (C2×C20).107(C2×C4), C2.4(C2×C22.F5), (C2×C10).61(C22×C4), (C22×C10).61(C2×C4), (C2×Dic5).183(C2×C4), SmallGroup(320,1092)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 330 in 118 conjugacy classes, 58 normal (30 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×2], C5, C8 [×4], C2×C4 [×2], C2×C4 [×12], C23, C10 [×3], C10 [×2], C42 [×4], C2×C8 [×4], C22×C4, C22×C4 [×2], Dic5 [×6], Dic5, C20 [×2], C20, C2×C10, C2×C10 [×2], C2×C10 [×2], C4×C8 [×2], C22⋊C8 [×2], C4⋊C8 [×2], C2×C42, C5⋊C8 [×4], C2×Dic5 [×4], C2×Dic5 [×4], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×2], C22×C10, C42.12C4, C4×Dic5 [×4], C2×C5⋊C8 [×4], C22×Dic5 [×2], C22×C20, C4×C5⋊C8 [×2], Dic5⋊C8 [×2], C23.2F5 [×2], C2×C4×Dic5, C20.34M4(2)
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, C2×C8 [×6], M4(2) [×2], C22×C4, C4○D4 [×2], F5, C42⋊C2, C22×C8, C2×M4(2), C2×F5 [×3], C42.12C4, D5⋊C8 [×2], C22.F5 [×2], C22×F5, C2×D5⋊C8, D10.C23, C2×C22.F5, C20.34M4(2)
Generators and relations
G = < a,b,c | a20=b8=c2=1, bab-1=a13, ac=ca, cbc=a10b5 >
►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 27 64 49 143 91 128 119)(2 24 73 42 144 88 137 112)(3 21 62 55 145 85 126 105)(4 38 71 48 146 82 135 118)(5 35 80 41 147 99 124 111)(6 32 69 54 148 96 133 104)(7 29 78 47 149 93 122 117)(8 26 67 60 150 90 131 110)(9 23 76 53 151 87 140 103)(10 40 65 46 152 84 129 116)(11 37 74 59 153 81 138 109)(12 34 63 52 154 98 127 102)(13 31 72 45 155 95 136 115)(14 28 61 58 156 92 125 108)(15 25 70 51 157 89 134 101)(16 22 79 44 158 86 123 114)(17 39 68 57 159 83 132 107)(18 36 77 50 160 100 121 120)(19 33 66 43 141 97 130 113)(20 30 75 56 142 94 139 106)
(21 95)(22 96)(23 97)(24 98)(25 99)(26 100)(27 81)(28 82)(29 83)(30 84)(31 85)(32 86)(33 87)(34 88)(35 89)(36 90)(37 91)(38 92)(39 93)(40 94)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)
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), (1,27,64,49,143,91,128,119)(2,24,73,42,144,88,137,112)(3,21,62,55,145,85,126,105)(4,38,71,48,146,82,135,118)(5,35,80,41,147,99,124,111)(6,32,69,54,148,96,133,104)(7,29,78,47,149,93,122,117)(8,26,67,60,150,90,131,110)(9,23,76,53,151,87,140,103)(10,40,65,46,152,84,129,116)(11,37,74,59,153,81,138,109)(12,34,63,52,154,98,127,102)(13,31,72,45,155,95,136,115)(14,28,61,58,156,92,125,108)(15,25,70,51,157,89,134,101)(16,22,79,44,158,86,123,114)(17,39,68,57,159,83,132,107)(18,36,77,50,160,100,121,120)(19,33,66,43,141,97,130,113)(20,30,75,56,142,94,139,106), (21,95)(22,96)(23,97)(24,98)(25,99)(26,100)(27,81)(28,82)(29,83)(30,84)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(37,91)(38,92)(39,93)(40,94)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120)>;
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), (1,27,64,49,143,91,128,119)(2,24,73,42,144,88,137,112)(3,21,62,55,145,85,126,105)(4,38,71,48,146,82,135,118)(5,35,80,41,147,99,124,111)(6,32,69,54,148,96,133,104)(7,29,78,47,149,93,122,117)(8,26,67,60,150,90,131,110)(9,23,76,53,151,87,140,103)(10,40,65,46,152,84,129,116)(11,37,74,59,153,81,138,109)(12,34,63,52,154,98,127,102)(13,31,72,45,155,95,136,115)(14,28,61,58,156,92,125,108)(15,25,70,51,157,89,134,101)(16,22,79,44,158,86,123,114)(17,39,68,57,159,83,132,107)(18,36,77,50,160,100,121,120)(19,33,66,43,141,97,130,113)(20,30,75,56,142,94,139,106), (21,95)(22,96)(23,97)(24,98)(25,99)(26,100)(27,81)(28,82)(29,83)(30,84)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(37,91)(38,92)(39,93)(40,94)(41,101)(42,102)(43,103)(44,104)(45,105)(46,106)(47,107)(48,108)(49,109)(50,110)(51,111)(52,112)(53,113)(54,114)(55,115)(56,116)(57,117)(58,118)(59,119)(60,120) );
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)], [(1,27,64,49,143,91,128,119),(2,24,73,42,144,88,137,112),(3,21,62,55,145,85,126,105),(4,38,71,48,146,82,135,118),(5,35,80,41,147,99,124,111),(6,32,69,54,148,96,133,104),(7,29,78,47,149,93,122,117),(8,26,67,60,150,90,131,110),(9,23,76,53,151,87,140,103),(10,40,65,46,152,84,129,116),(11,37,74,59,153,81,138,109),(12,34,63,52,154,98,127,102),(13,31,72,45,155,95,136,115),(14,28,61,58,156,92,125,108),(15,25,70,51,157,89,134,101),(16,22,79,44,158,86,123,114),(17,39,68,57,159,83,132,107),(18,36,77,50,160,100,121,120),(19,33,66,43,141,97,130,113),(20,30,75,56,142,94,139,106)], [(21,95),(22,96),(23,97),(24,98),(25,99),(26,100),(27,81),(28,82),(29,83),(30,84),(31,85),(32,86),(33,87),(34,88),(35,89),(36,90),(37,91),(38,92),(39,93),(40,94),(41,101),(42,102),(43,103),(44,104),(45,105),(46,106),(47,107),(48,108),(49,109),(50,110),(51,111),(52,112),(53,113),(54,114),(55,115),(56,116),(57,117),(58,118),(59,119),(60,120)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 40 | 0 | 0 |
| 0 | 0 | 8 | 40 | 0 | 0 |
| 0 | 0 | 32 | 7 | 0 | 35 |
| 0 | 0 | 33 | 14 | 7 | 34 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 21 | 39 | 0 |
| 0 | 0 | 27 | 20 | 0 | 39 |
| 0 | 0 | 34 | 29 | 38 | 20 |
| 0 | 0 | 2 | 20 | 14 | 21 |
| 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 | 3 | 21 | 40 | 0 |
| 0 | 0 | 27 | 20 | 0 | 40 |
G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,7,8,32,33,0,0,40,40,7,14,0,0,0,0,0,7,0,0,0,0,35,34],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,3,27,34,2,0,0,21,20,29,20,0,0,39,0,38,14,0,0,0,39,20,21],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,3,27,0,0,0,1,21,20,0,0,0,0,40,0,0,0,0,0,0,40] >;
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 4O | 4P | 4Q | 4R | 5 | 8A | ··· | 8P | 10A | ··· | 10G | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 4 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | C4○D4 | M4(2) | F5 | C2×F5 | C2×F5 | C22.F5 | D5⋊C8 | D10.C23 |
| kernel | C20.34M4(2) | C4×C5⋊C8 | Dic5⋊C8 | C23.2F5 | C2×C4×Dic5 | C4×Dic5 | C22×Dic5 | C22×C20 | C2×Dic5 | Dic5 | C20 | C22×C4 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 4 | 2 | 2 | 16 | 4 | 4 | 1 | 2 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_{20}._{34}M_{4(2)} % in TeX
G:=Group("C20.34M4(2)"); // GroupNames label
G:=SmallGroup(320,1092);
// by ID
G=gap.SmallGroup(320,1092);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,253,268,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^13,a*c=c*a,c*b*c=a^10*b^5>;
// generators/relations
×
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ℤ
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