metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊7M4(2), C42.206D10, C5⋊2C8⋊18D4, C5⋊7(C8⋊6D4), (C4×D4).6D5, (D4×C20).7C2, C20⋊3C8⋊22C2, C4.216(D4×D5), C4⋊C4.8Dic5, C2.9(D4×Dic5), (D4×C10).22C4, C10.122(C4×D4), C20.375(C2×D4), C4⋊1(C4.Dic5), (C2×D4).6Dic5, C10.64(C8○D4), C20.55D4⋊3C2, (C4×C20).83C22, C22⋊C4.5Dic5, C20.308(C4○D4), C23.9(C2×Dic5), (C2×C20).850C23, (C22×C4).115D10, C10.75(C2×M4(2)), C2.7(D4.Dic5), C4.135(D4⋊2D5), (C22×C20).100C22, C22.46(C22×Dic5), (C4×C5⋊2C8)⋊7C2, (C5×C4⋊C4).25C4, (C2×C4.Dic5)⋊5C2, (C2×C20).337(C2×C4), C2.9(C2×C4.Dic5), (C5×C22⋊C4).14C4, (C2×C4).20(C2×Dic5), (C2×C4).792(C22×D5), (C2×C10).288(C22×C4), (C22×C10).129(C2×C4), (C2×C5⋊2C8).205C22, SmallGroup(320,639)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20⋊7M4(2)
G = < a,b,c | a20=b8=c2=1, bab-1=a-1, cac=a11, cbc=b5 >
Subgroups: 286 in 122 conjugacy classes, 61 normal (33 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C20, C20, C20, C2×C10, C2×C10, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C2×M4(2), C5⋊2C8, C5⋊2C8, C2×C20, C2×C20, C2×C20, C5×D4, C22×C10, C8⋊6D4, C2×C5⋊2C8, C2×C5⋊2C8, C4.Dic5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, C4×C5⋊2C8, C20⋊3C8, C20.55D4, C2×C4.Dic5, D4×C20, C20⋊7M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, M4(2), C22×C4, C2×D4, C4○D4, Dic5, D10, C4×D4, C2×M4(2), C8○D4, C2×Dic5, C22×D5, C8⋊6D4, C4.Dic5, D4×D5, D4⋊2D5, C22×Dic5, C2×C4.Dic5, D4×Dic5, D4.Dic5, C20⋊7M4(2)
►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)
(1 131 110 34 69 99 60 153)(2 130 111 33 70 98 41 152)(3 129 112 32 71 97 42 151)(4 128 113 31 72 96 43 150)(5 127 114 30 73 95 44 149)(6 126 115 29 74 94 45 148)(7 125 116 28 75 93 46 147)(8 124 117 27 76 92 47 146)(9 123 118 26 77 91 48 145)(10 122 119 25 78 90 49 144)(11 121 120 24 79 89 50 143)(12 140 101 23 80 88 51 142)(13 139 102 22 61 87 52 141)(14 138 103 21 62 86 53 160)(15 137 104 40 63 85 54 159)(16 136 105 39 64 84 55 158)(17 135 106 38 65 83 56 157)(18 134 107 37 66 82 57 156)(19 133 108 36 67 81 58 155)(20 132 109 35 68 100 59 154)
(1 74)(2 65)(3 76)(4 67)(5 78)(6 69)(7 80)(8 71)(9 62)(10 73)(11 64)(12 75)(13 66)(14 77)(15 68)(16 79)(17 70)(18 61)(19 72)(20 63)(21 26)(22 37)(23 28)(24 39)(25 30)(27 32)(29 34)(31 36)(33 38)(35 40)(41 106)(42 117)(43 108)(44 119)(45 110)(46 101)(47 112)(48 103)(49 114)(50 105)(51 116)(52 107)(53 118)(54 109)(55 120)(56 111)(57 102)(58 113)(59 104)(60 115)(81 96)(82 87)(83 98)(84 89)(85 100)(86 91)(88 93)(90 95)(92 97)(94 99)(121 136)(122 127)(123 138)(124 129)(125 140)(126 131)(128 133)(130 135)(132 137)(134 139)(141 156)(142 147)(143 158)(144 149)(145 160)(146 151)(148 153)(150 155)(152 157)(154 159)
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), (1,131,110,34,69,99,60,153)(2,130,111,33,70,98,41,152)(3,129,112,32,71,97,42,151)(4,128,113,31,72,96,43,150)(5,127,114,30,73,95,44,149)(6,126,115,29,74,94,45,148)(7,125,116,28,75,93,46,147)(8,124,117,27,76,92,47,146)(9,123,118,26,77,91,48,145)(10,122,119,25,78,90,49,144)(11,121,120,24,79,89,50,143)(12,140,101,23,80,88,51,142)(13,139,102,22,61,87,52,141)(14,138,103,21,62,86,53,160)(15,137,104,40,63,85,54,159)(16,136,105,39,64,84,55,158)(17,135,106,38,65,83,56,157)(18,134,107,37,66,82,57,156)(19,133,108,36,67,81,58,155)(20,132,109,35,68,100,59,154), (1,74)(2,65)(3,76)(4,67)(5,78)(6,69)(7,80)(8,71)(9,62)(10,73)(11,64)(12,75)(13,66)(14,77)(15,68)(16,79)(17,70)(18,61)(19,72)(20,63)(21,26)(22,37)(23,28)(24,39)(25,30)(27,32)(29,34)(31,36)(33,38)(35,40)(41,106)(42,117)(43,108)(44,119)(45,110)(46,101)(47,112)(48,103)(49,114)(50,105)(51,116)(52,107)(53,118)(54,109)(55,120)(56,111)(57,102)(58,113)(59,104)(60,115)(81,96)(82,87)(83,98)(84,89)(85,100)(86,91)(88,93)(90,95)(92,97)(94,99)(121,136)(122,127)(123,138)(124,129)(125,140)(126,131)(128,133)(130,135)(132,137)(134,139)(141,156)(142,147)(143,158)(144,149)(145,160)(146,151)(148,153)(150,155)(152,157)(154,159)>;
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), (1,131,110,34,69,99,60,153)(2,130,111,33,70,98,41,152)(3,129,112,32,71,97,42,151)(4,128,113,31,72,96,43,150)(5,127,114,30,73,95,44,149)(6,126,115,29,74,94,45,148)(7,125,116,28,75,93,46,147)(8,124,117,27,76,92,47,146)(9,123,118,26,77,91,48,145)(10,122,119,25,78,90,49,144)(11,121,120,24,79,89,50,143)(12,140,101,23,80,88,51,142)(13,139,102,22,61,87,52,141)(14,138,103,21,62,86,53,160)(15,137,104,40,63,85,54,159)(16,136,105,39,64,84,55,158)(17,135,106,38,65,83,56,157)(18,134,107,37,66,82,57,156)(19,133,108,36,67,81,58,155)(20,132,109,35,68,100,59,154), (1,74)(2,65)(3,76)(4,67)(5,78)(6,69)(7,80)(8,71)(9,62)(10,73)(11,64)(12,75)(13,66)(14,77)(15,68)(16,79)(17,70)(18,61)(19,72)(20,63)(21,26)(22,37)(23,28)(24,39)(25,30)(27,32)(29,34)(31,36)(33,38)(35,40)(41,106)(42,117)(43,108)(44,119)(45,110)(46,101)(47,112)(48,103)(49,114)(50,105)(51,116)(52,107)(53,118)(54,109)(55,120)(56,111)(57,102)(58,113)(59,104)(60,115)(81,96)(82,87)(83,98)(84,89)(85,100)(86,91)(88,93)(90,95)(92,97)(94,99)(121,136)(122,127)(123,138)(124,129)(125,140)(126,131)(128,133)(130,135)(132,137)(134,139)(141,156)(142,147)(143,158)(144,149)(145,160)(146,151)(148,153)(150,155)(152,157)(154,159) );
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)], [(1,131,110,34,69,99,60,153),(2,130,111,33,70,98,41,152),(3,129,112,32,71,97,42,151),(4,128,113,31,72,96,43,150),(5,127,114,30,73,95,44,149),(6,126,115,29,74,94,45,148),(7,125,116,28,75,93,46,147),(8,124,117,27,76,92,47,146),(9,123,118,26,77,91,48,145),(10,122,119,25,78,90,49,144),(11,121,120,24,79,89,50,143),(12,140,101,23,80,88,51,142),(13,139,102,22,61,87,52,141),(14,138,103,21,62,86,53,160),(15,137,104,40,63,85,54,159),(16,136,105,39,64,84,55,158),(17,135,106,38,65,83,56,157),(18,134,107,37,66,82,57,156),(19,133,108,36,67,81,58,155),(20,132,109,35,68,100,59,154)], [(1,74),(2,65),(3,76),(4,67),(5,78),(6,69),(7,80),(8,71),(9,62),(10,73),(11,64),(12,75),(13,66),(14,77),(15,68),(16,79),(17,70),(18,61),(19,72),(20,63),(21,26),(22,37),(23,28),(24,39),(25,30),(27,32),(29,34),(31,36),(33,38),(35,40),(41,106),(42,117),(43,108),(44,119),(45,110),(46,101),(47,112),(48,103),(49,114),(50,105),(51,116),(52,107),(53,118),(54,109),(55,120),(56,111),(57,102),(58,113),(59,104),(60,115),(81,96),(82,87),(83,98),(84,89),(85,100),(86,91),(88,93),(90,95),(92,97),(94,99),(121,136),(122,127),(123,138),(124,129),(125,140),(126,131),(128,133),(130,135),(132,137),(134,139),(141,156),(142,147),(143,158),(144,149),(145,160),(146,151),(148,153),(150,155),(152,157),(154,159)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | + | - | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D5 | M4(2) | C4○D4 | D10 | Dic5 | Dic5 | D10 | Dic5 | C8○D4 | C4.Dic5 | D4×D5 | D4⋊2D5 | D4.Dic5 |
| kernel | C20⋊7M4(2) | C4×C5⋊2C8 | C20⋊3C8 | C20.55D4 | C2×C4.Dic5 | D4×C20 | C5×C22⋊C4 | C5×C4⋊C4 | D4×C10 | C5⋊2C8 | C4×D4 | C20 | C20 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C10 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 2 | 4 | 2 | 4 | 16 | 2 | 2 | 4 |
Matrix representation of C20⋊7M4(2) ►in GL4(𝔽41) generated by
| 10 | 0 | 0 | 0 |
| 14 | 37 | 0 | 0 |
| 0 | 0 | 40 | 5 |
| 0 | 0 | 16 | 1 |
| 30 | 11 | 0 | 0 |
| 1 | 11 | 0 | 0 |
| 0 | 0 | 38 | 0 |
| 0 | 0 | 7 | 3 |
| 40 | 0 | 0 | 0 |
| 39 | 1 | 0 | 0 |
| 0 | 0 | 40 | 5 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [10,14,0,0,0,37,0,0,0,0,40,16,0,0,5,1],[30,1,0,0,11,11,0,0,0,0,38,7,0,0,0,3],[40,39,0,0,0,1,0,0,0,0,40,0,0,0,5,1] >;
C20⋊7M4(2) in GAP, Magma, Sage, TeX
C_{20}\rtimes_7M_4(2) % in TeX
G:=Group("C20:7M4(2)"); // GroupNames label
G:=SmallGroup(320,639);
// by ID
G=gap.SmallGroup(320,639);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,758,219,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^-1,c*a*c=a^11,c*b*c=b^5>;
// generators/relations