metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊13M4(2), C42.269D10, C42.12Dic5, C20⋊3C8⋊4C2, (C4×C20).28C4, C4.80(C2×D20), (C2×C4).87D20, C20.62(C4⋊C4), C20.83(C2×Q8), (C2×C20).59Q8, (C2×C20).396D4, C20.300(C2×D4), C4⋊2(C4.Dic5), (C2×C42).10D5, (C22×C20).51C4, C5⋊5(C4⋊M4(2)), C4.14(C4⋊Dic5), (C2×C4).44Dic10, C4.48(C2×Dic10), (C4×C20).330C22, (C2×C20).842C23, (C22×C4).416D10, C10.71(C2×M4(2)), C23.25(C2×Dic5), (C22×C4).15Dic5, C22.12(C4⋊Dic5), (C22×C20).535C22, C22.34(C22×Dic5), (C2×C4×C20).18C2, C10.46(C2×C4⋊C4), C2.4(C2×C4⋊Dic5), (C2×C10).71(C4⋊C4), (C2×C20).468(C2×C4), C2.6(C2×C4.Dic5), (C2×C4).58(C2×Dic5), (C2×C4.Dic5).3C2, (C2×C4).784(C22×D5), (C22×C10).198(C2×C4), (C2×C10).272(C22×C4), (C2×C5⋊2C8).202C22, SmallGroup(320,551)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20⋊13M4(2)
G = < a,b,c | a20=b8=c2=1, bab-1=a-1, ac=ca, cbc=b5 >
Subgroups: 270 in 126 conjugacy classes, 79 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C10, C10, C10, C42, C42, C2×C8, M4(2), C22×C4, C22×C4, C20, C20, C20, C2×C10, C2×C10, C2×C10, C4⋊C8, C2×C42, C2×M4(2), C5⋊2C8, C2×C20, C2×C20, C2×C20, C22×C10, C4⋊M4(2), C2×C5⋊2C8, C4.Dic5, C4×C20, C4×C20, C22×C20, C22×C20, C20⋊3C8, C2×C4.Dic5, C2×C4×C20, C20⋊13M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C4⋊C4, M4(2), C22×C4, C2×D4, C2×Q8, Dic5, D10, C2×C4⋊C4, C2×M4(2), Dic10, D20, C2×Dic5, C22×D5, C4⋊M4(2), C4.Dic5, C4⋊Dic5, C2×Dic10, C2×D20, C22×Dic5, C2×C4.Dic5, C2×C4⋊Dic5, C20⋊13M4(2)
►On 160 points
(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 69 102 147 128 56 39 92)(2 68 103 146 129 55 40 91)(3 67 104 145 130 54 21 90)(4 66 105 144 131 53 22 89)(5 65 106 143 132 52 23 88)(6 64 107 142 133 51 24 87)(7 63 108 141 134 50 25 86)(8 62 109 160 135 49 26 85)(9 61 110 159 136 48 27 84)(10 80 111 158 137 47 28 83)(11 79 112 157 138 46 29 82)(12 78 113 156 139 45 30 81)(13 77 114 155 140 44 31 100)(14 76 115 154 121 43 32 99)(15 75 116 153 122 42 33 98)(16 74 117 152 123 41 34 97)(17 73 118 151 124 60 35 96)(18 72 119 150 125 59 36 95)(19 71 120 149 126 58 37 94)(20 70 101 148 127 57 38 93)
(41 74)(42 75)(43 76)(44 77)(45 78)(46 79)(47 80)(48 61)(49 62)(50 63)(51 64)(52 65)(53 66)(54 67)(55 68)(56 69)(57 70)(58 71)(59 72)(60 73)(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), (1,69,102,147,128,56,39,92)(2,68,103,146,129,55,40,91)(3,67,104,145,130,54,21,90)(4,66,105,144,131,53,22,89)(5,65,106,143,132,52,23,88)(6,64,107,142,133,51,24,87)(7,63,108,141,134,50,25,86)(8,62,109,160,135,49,26,85)(9,61,110,159,136,48,27,84)(10,80,111,158,137,47,28,83)(11,79,112,157,138,46,29,82)(12,78,113,156,139,45,30,81)(13,77,114,155,140,44,31,100)(14,76,115,154,121,43,32,99)(15,75,116,153,122,42,33,98)(16,74,117,152,123,41,34,97)(17,73,118,151,124,60,35,96)(18,72,119,150,125,59,36,95)(19,71,120,149,126,58,37,94)(20,70,101,148,127,57,38,93), (41,74)(42,75)(43,76)(44,77)(45,78)(46,79)(47,80)(48,61)(49,62)(50,63)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(57,70)(58,71)(59,72)(60,73)(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), (1,69,102,147,128,56,39,92)(2,68,103,146,129,55,40,91)(3,67,104,145,130,54,21,90)(4,66,105,144,131,53,22,89)(5,65,106,143,132,52,23,88)(6,64,107,142,133,51,24,87)(7,63,108,141,134,50,25,86)(8,62,109,160,135,49,26,85)(9,61,110,159,136,48,27,84)(10,80,111,158,137,47,28,83)(11,79,112,157,138,46,29,82)(12,78,113,156,139,45,30,81)(13,77,114,155,140,44,31,100)(14,76,115,154,121,43,32,99)(15,75,116,153,122,42,33,98)(16,74,117,152,123,41,34,97)(17,73,118,151,124,60,35,96)(18,72,119,150,125,59,36,95)(19,71,120,149,126,58,37,94)(20,70,101,148,127,57,38,93), (41,74)(42,75)(43,76)(44,77)(45,78)(46,79)(47,80)(48,61)(49,62)(50,63)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(57,70)(58,71)(59,72)(60,73)(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)], [(1,69,102,147,128,56,39,92),(2,68,103,146,129,55,40,91),(3,67,104,145,130,54,21,90),(4,66,105,144,131,53,22,89),(5,65,106,143,132,52,23,88),(6,64,107,142,133,51,24,87),(7,63,108,141,134,50,25,86),(8,62,109,160,135,49,26,85),(9,61,110,159,136,48,27,84),(10,80,111,158,137,47,28,83),(11,79,112,157,138,46,29,82),(12,78,113,156,139,45,30,81),(13,77,114,155,140,44,31,100),(14,76,115,154,121,43,32,99),(15,75,116,153,122,42,33,98),(16,74,117,152,123,41,34,97),(17,73,118,151,124,60,35,96),(18,72,119,150,125,59,36,95),(19,71,120,149,126,58,37,94),(20,70,101,148,127,57,38,93)], [(41,74),(42,75),(43,76),(44,77),(45,78),(46,79),(47,80),(48,61),(49,62),(50,63),(51,64),(52,65),(53,66),(54,67),(55,68),(56,69),(57,70),(58,71),(59,72),(60,73),(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)]])
92 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10N | 20A | ··· | 20AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 20 | ··· | 20 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | + | - | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | Q8 | D5 | M4(2) | Dic5 | D10 | Dic5 | D10 | Dic10 | D20 | C4.Dic5 |
| kernel | C20⋊13M4(2) | C20⋊3C8 | C2×C4.Dic5 | C2×C4×C20 | C4×C20 | C22×C20 | C2×C20 | C2×C20 | C2×C42 | C20 | C42 | C42 | C22×C4 | C22×C4 | C2×C4 | C2×C4 | C4 |
| # reps | 1 | 4 | 2 | 1 | 4 | 4 | 2 | 2 | 2 | 8 | 4 | 4 | 4 | 2 | 8 | 8 | 32 |
Matrix representation of C20⋊13M4(2) ►in GL4(𝔽41) generated by
| 39 | 0 | 0 | 0 |
| 0 | 20 | 0 | 0 |
| 0 | 0 | 16 | 2 |
| 0 | 0 | 39 | 28 |
| 0 | 1 | 0 | 0 |
| 32 | 0 | 0 | 0 |
| 0 | 0 | 5 | 3 |
| 0 | 0 | 33 | 36 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [39,0,0,0,0,20,0,0,0,0,16,39,0,0,2,28],[0,32,0,0,1,0,0,0,0,0,5,33,0,0,3,36],[1,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1] >;
C20⋊13M4(2) in GAP, Magma, Sage, TeX
C_{20}\rtimes_{13}M_4(2) % in TeX
G:=Group("C20:13M4(2)"); // GroupNames label
G:=SmallGroup(320,551);
// by ID
G=gap.SmallGroup(320,551);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,253,120,758,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^-1,a*c=c*a,c*b*c=b^5>;
// generators/relations