metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.30M4(2), C20⋊C8⋊16C2, C23.46(C2×F5), (C22×C4).23F5, (C22×C20).25C4, C5⋊4(C42.6C4), (C4×Dic5).33C4, (C2×C10).9M4(2), C22.6(C4.F5), C4.8(C22.F5), C10.C42⋊8C2, C10.29(C2×M4(2)), C23.2F5.5C2, Dic5.32(C4○D4), C22.88(C22×F5), (C22×Dic5).35C4, C10.16(C42⋊C2), (C2×Dic5).350C23, (C4×Dic5).349C22, (C22×Dic5).278C22, C2.17(D10.C23), (C2×C5⋊C8).9C22, C2.12(C2×C4.F5), (C2×C4).111(C2×F5), (C2×C4×Dic5).48C2, (C2×C20).132(C2×C4), C2.8(C2×C22.F5), (C2×C10).66(C22×C4), (C22×C10).66(C2×C4), (C2×Dic5).188(C2×C4), SmallGroup(320,1097)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 330 in 110 conjugacy classes, 50 normal (30 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×2], C5, C8 [×4], C2×C4 [×2], C2×C4 [×10], C23, C10 [×3], C10 [×2], C42 [×4], C2×C8 [×4], C22×C4, C22×C4 [×2], Dic5 [×2], Dic5 [×3], C20 [×2], C20, C2×C10, C2×C10 [×2], C2×C10 [×2], C8⋊C4 [×2], C22⋊C8 [×2], C4⋊C8 [×2], C2×C42, C5⋊C8 [×4], C2×Dic5 [×4], C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×2], C22×C10, C42.6C4, C4×Dic5 [×4], C2×C5⋊C8 [×4], C22×Dic5 [×2], C22×C20, C20⋊C8 [×2], C10.C42 [×2], C23.2F5 [×2], C2×C4×Dic5, C20.30M4(2)
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, M4(2) [×4], C22×C4, C4○D4 [×2], F5, C42⋊C2, C2×M4(2) [×2], C2×F5 [×3], C42.6C4, C4.F5 [×2], C22.F5 [×2], C22×F5, C2×C4.F5, D10.C23, C2×C22.F5, C20.30M4(2)
Generators and relations
G = < a,b,c | a20=b8=1, c2=a10, bab-1=a3, ac=ca, cbc-1=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 54 143 37 129 107 94 67)(2 41 152 40 130 114 83 70)(3 48 141 23 131 101 92 73)(4 55 150 26 132 108 81 76)(5 42 159 29 133 115 90 79)(6 49 148 32 134 102 99 62)(7 56 157 35 135 109 88 65)(8 43 146 38 136 116 97 68)(9 50 155 21 137 103 86 71)(10 57 144 24 138 110 95 74)(11 44 153 27 139 117 84 77)(12 51 142 30 140 104 93 80)(13 58 151 33 121 111 82 63)(14 45 160 36 122 118 91 66)(15 52 149 39 123 105 100 69)(16 59 158 22 124 112 89 72)(17 46 147 25 125 119 98 75)(18 53 156 28 126 106 87 78)(19 60 145 31 127 113 96 61)(20 47 154 34 128 120 85 64)
(1 6 11 16)(2 7 12 17)(3 8 13 18)(4 9 14 19)(5 10 15 20)(21 66 31 76)(22 67 32 77)(23 68 33 78)(24 69 34 79)(25 70 35 80)(26 71 36 61)(27 72 37 62)(28 73 38 63)(29 74 39 64)(30 75 40 65)(41 109 51 119)(42 110 52 120)(43 111 53 101)(44 112 54 102)(45 113 55 103)(46 114 56 104)(47 115 57 105)(48 116 58 106)(49 117 59 107)(50 118 60 108)(81 86 91 96)(82 87 92 97)(83 88 93 98)(84 89 94 99)(85 90 95 100)(121 126 131 136)(122 127 132 137)(123 128 133 138)(124 129 134 139)(125 130 135 140)(141 146 151 156)(142 147 152 157)(143 148 153 158)(144 149 154 159)(145 150 155 160)
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), (1,54,143,37,129,107,94,67)(2,41,152,40,130,114,83,70)(3,48,141,23,131,101,92,73)(4,55,150,26,132,108,81,76)(5,42,159,29,133,115,90,79)(6,49,148,32,134,102,99,62)(7,56,157,35,135,109,88,65)(8,43,146,38,136,116,97,68)(9,50,155,21,137,103,86,71)(10,57,144,24,138,110,95,74)(11,44,153,27,139,117,84,77)(12,51,142,30,140,104,93,80)(13,58,151,33,121,111,82,63)(14,45,160,36,122,118,91,66)(15,52,149,39,123,105,100,69)(16,59,158,22,124,112,89,72)(17,46,147,25,125,119,98,75)(18,53,156,28,126,106,87,78)(19,60,145,31,127,113,96,61)(20,47,154,34,128,120,85,64), (1,6,11,16)(2,7,12,17)(3,8,13,18)(4,9,14,19)(5,10,15,20)(21,66,31,76)(22,67,32,77)(23,68,33,78)(24,69,34,79)(25,70,35,80)(26,71,36,61)(27,72,37,62)(28,73,38,63)(29,74,39,64)(30,75,40,65)(41,109,51,119)(42,110,52,120)(43,111,53,101)(44,112,54,102)(45,113,55,103)(46,114,56,104)(47,115,57,105)(48,116,58,106)(49,117,59,107)(50,118,60,108)(81,86,91,96)(82,87,92,97)(83,88,93,98)(84,89,94,99)(85,90,95,100)(121,126,131,136)(122,127,132,137)(123,128,133,138)(124,129,134,139)(125,130,135,140)(141,146,151,156)(142,147,152,157)(143,148,153,158)(144,149,154,159)(145,150,155,160)>;
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), (1,54,143,37,129,107,94,67)(2,41,152,40,130,114,83,70)(3,48,141,23,131,101,92,73)(4,55,150,26,132,108,81,76)(5,42,159,29,133,115,90,79)(6,49,148,32,134,102,99,62)(7,56,157,35,135,109,88,65)(8,43,146,38,136,116,97,68)(9,50,155,21,137,103,86,71)(10,57,144,24,138,110,95,74)(11,44,153,27,139,117,84,77)(12,51,142,30,140,104,93,80)(13,58,151,33,121,111,82,63)(14,45,160,36,122,118,91,66)(15,52,149,39,123,105,100,69)(16,59,158,22,124,112,89,72)(17,46,147,25,125,119,98,75)(18,53,156,28,126,106,87,78)(19,60,145,31,127,113,96,61)(20,47,154,34,128,120,85,64), (1,6,11,16)(2,7,12,17)(3,8,13,18)(4,9,14,19)(5,10,15,20)(21,66,31,76)(22,67,32,77)(23,68,33,78)(24,69,34,79)(25,70,35,80)(26,71,36,61)(27,72,37,62)(28,73,38,63)(29,74,39,64)(30,75,40,65)(41,109,51,119)(42,110,52,120)(43,111,53,101)(44,112,54,102)(45,113,55,103)(46,114,56,104)(47,115,57,105)(48,116,58,106)(49,117,59,107)(50,118,60,108)(81,86,91,96)(82,87,92,97)(83,88,93,98)(84,89,94,99)(85,90,95,100)(121,126,131,136)(122,127,132,137)(123,128,133,138)(124,129,134,139)(125,130,135,140)(141,146,151,156)(142,147,152,157)(143,148,153,158)(144,149,154,159)(145,150,155,160) );
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)], [(1,54,143,37,129,107,94,67),(2,41,152,40,130,114,83,70),(3,48,141,23,131,101,92,73),(4,55,150,26,132,108,81,76),(5,42,159,29,133,115,90,79),(6,49,148,32,134,102,99,62),(7,56,157,35,135,109,88,65),(8,43,146,38,136,116,97,68),(9,50,155,21,137,103,86,71),(10,57,144,24,138,110,95,74),(11,44,153,27,139,117,84,77),(12,51,142,30,140,104,93,80),(13,58,151,33,121,111,82,63),(14,45,160,36,122,118,91,66),(15,52,149,39,123,105,100,69),(16,59,158,22,124,112,89,72),(17,46,147,25,125,119,98,75),(18,53,156,28,126,106,87,78),(19,60,145,31,127,113,96,61),(20,47,154,34,128,120,85,64)], [(1,6,11,16),(2,7,12,17),(3,8,13,18),(4,9,14,19),(5,10,15,20),(21,66,31,76),(22,67,32,77),(23,68,33,78),(24,69,34,79),(25,70,35,80),(26,71,36,61),(27,72,37,62),(28,73,38,63),(29,74,39,64),(30,75,40,65),(41,109,51,119),(42,110,52,120),(43,111,53,101),(44,112,54,102),(45,113,55,103),(46,114,56,104),(47,115,57,105),(48,116,58,106),(49,117,59,107),(50,118,60,108),(81,86,91,96),(82,87,92,97),(83,88,93,98),(84,89,94,99),(85,90,95,100),(121,126,131,136),(122,127,132,137),(123,128,133,138),(124,129,134,139),(125,130,135,140),(141,146,151,156),(142,147,152,157),(143,148,153,158),(144,149,154,159),(145,150,155,160)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 1 | 39 | 0 | 0 | 0 | 0 |
| 1 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 9 | 19 | 0 | 0 |
| 0 | 0 | 16 | 6 | 9 | 19 |
| 0 | 0 | 11 | 5 | 22 | 19 |
| 31 | 10 | 0 | 0 | 0 | 0 |
| 36 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 0 | 23 | 0 |
| 0 | 0 | 31 | 34 | 0 | 23 |
| 0 | 0 | 13 | 27 | 20 | 0 |
| 0 | 0 | 23 | 37 | 10 | 7 |
| 40 | 2 | 0 | 0 | 0 | 0 |
| 40 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
G:=sub<GL(6,GF(41))| [1,1,0,0,0,0,39,40,0,0,0,0,0,0,0,9,16,11,0,0,32,19,6,5,0,0,0,0,9,22,0,0,0,0,19,19],[31,36,0,0,0,0,10,10,0,0,0,0,0,0,21,31,13,23,0,0,0,34,27,37,0,0,23,0,20,10,0,0,0,23,0,7],[40,40,0,0,0,0,2,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9] >;
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 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4○D4 | M4(2) | M4(2) | F5 | C2×F5 | C2×F5 | C22.F5 | C4.F5 | D10.C23 |
| kernel | C20.30M4(2) | C20⋊C8 | C10.C42 | C23.2F5 | C2×C4×Dic5 | C4×Dic5 | C22×Dic5 | C22×C20 | Dic5 | C20 | C2×C10 | C22×C4 | C2×C4 | C23 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 4 | 2 | 2 | 4 | 4 | 4 | 1 | 2 | 1 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_{20}._{30}M_{4(2)} % in TeX
G:=Group("C20.30M4(2)"); // GroupNames label
G:=SmallGroup(320,1097);
// by ID
G=gap.SmallGroup(320,1097);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,120,1094,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c|a^20=b^8=1,c^2=a^10,b*a*b^-1=a^3,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations
×
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ℤ
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