metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊1M4(2), C5⋊C8⋊2D4, C4⋊C4.6F5, C5⋊2(C8⋊6D4), C20⋊C8⋊4C2, C4⋊1(C4.F5), C2.12(D4×F5), C10.10(C4×D4), (C2×D20).12C4, D10⋊C8⋊10C2, C2.5(Q8.F5), D10⋊C4.3C4, C10.21(C8○D4), Dic5.71(C2×D4), D20⋊8C4.18C2, C10.14(C2×M4(2)), Dic5.56(C4○D4), C22.77(C22×F5), (C4×Dic5).192C22, (C2×Dic5).332C23, (C4×C5⋊C8)⋊4C2, (C5×C4⋊C4).9C4, C2.9(C2×C4.F5), (C2×C4.F5)⋊12C2, (C2×C4).26(C2×F5), (C2×C20).83(C2×C4), (C2×C5⋊C8).29C22, (C2×C4×D5).277C22, (C2×C10).43(C22×C4), (C2×Dic5).58(C2×C4), (C22×D5).49(C2×C4), SmallGroup(320,1043)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20⋊M4(2)
G = < a,b,c | a20=b8=c2=1, bab-1=a7, cac=a-1, cbc=b5 >
Subgroups: 474 in 122 conjugacy classes, 48 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, Dic5, Dic5, C20, C20, D10, C2×C10, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C2×M4(2), C5⋊C8, C5⋊C8, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C8⋊6D4, C4×Dic5, D10⋊C4, C5×C4⋊C4, C4.F5, C2×C5⋊C8, C2×C5⋊C8, C2×C4×D5, C2×D20, C4×C5⋊C8, C20⋊C8, D10⋊C8, D20⋊8C4, C2×C4.F5, C20⋊M4(2)
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, M4(2), C22×C4, C2×D4, C4○D4, F5, C4×D4, C2×M4(2), C8○D4, C2×F5, C8⋊6D4, C4.F5, C22×F5, C2×C4.F5, D4×F5, Q8.F5, C20⋊M4(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 98 27 47 146 62 111 133)(2 81 36 54 147 65 120 140)(3 84 25 41 148 68 109 127)(4 87 34 48 149 71 118 134)(5 90 23 55 150 74 107 121)(6 93 32 42 151 77 116 128)(7 96 21 49 152 80 105 135)(8 99 30 56 153 63 114 122)(9 82 39 43 154 66 103 129)(10 85 28 50 155 69 112 136)(11 88 37 57 156 72 101 123)(12 91 26 44 157 75 110 130)(13 94 35 51 158 78 119 137)(14 97 24 58 159 61 108 124)(15 100 33 45 160 64 117 131)(16 83 22 52 141 67 106 138)(17 86 31 59 142 70 115 125)(18 89 40 46 143 73 104 132)(19 92 29 53 144 76 113 139)(20 95 38 60 145 79 102 126)
(1 151)(2 150)(3 149)(4 148)(5 147)(6 146)(7 145)(8 144)(9 143)(10 142)(11 141)(12 160)(13 159)(14 158)(15 157)(16 156)(17 155)(18 154)(19 153)(20 152)(21 102)(22 101)(23 120)(24 119)(25 118)(26 117)(27 116)(28 115)(29 114)(30 113)(31 112)(32 111)(33 110)(34 109)(35 108)(36 107)(37 106)(38 105)(39 104)(40 103)(41 48)(42 47)(43 46)(44 45)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(61 78)(62 77)(63 76)(64 75)(65 74)(66 73)(67 72)(68 71)(69 70)(79 80)(81 90)(82 89)(83 88)(84 87)(85 86)(91 100)(92 99)(93 98)(94 97)(95 96)(121 140)(122 139)(123 138)(124 137)(125 136)(126 135)(127 134)(128 133)(129 132)(130 131)
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), (1,98,27,47,146,62,111,133)(2,81,36,54,147,65,120,140)(3,84,25,41,148,68,109,127)(4,87,34,48,149,71,118,134)(5,90,23,55,150,74,107,121)(6,93,32,42,151,77,116,128)(7,96,21,49,152,80,105,135)(8,99,30,56,153,63,114,122)(9,82,39,43,154,66,103,129)(10,85,28,50,155,69,112,136)(11,88,37,57,156,72,101,123)(12,91,26,44,157,75,110,130)(13,94,35,51,158,78,119,137)(14,97,24,58,159,61,108,124)(15,100,33,45,160,64,117,131)(16,83,22,52,141,67,106,138)(17,86,31,59,142,70,115,125)(18,89,40,46,143,73,104,132)(19,92,29,53,144,76,113,139)(20,95,38,60,145,79,102,126), (1,151)(2,150)(3,149)(4,148)(5,147)(6,146)(7,145)(8,144)(9,143)(10,142)(11,141)(12,160)(13,159)(14,158)(15,157)(16,156)(17,155)(18,154)(19,153)(20,152)(21,102)(22,101)(23,120)(24,119)(25,118)(26,117)(27,116)(28,115)(29,114)(30,113)(31,112)(32,111)(33,110)(34,109)(35,108)(36,107)(37,106)(38,105)(39,104)(40,103)(41,48)(42,47)(43,46)(44,45)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)(79,80)(81,90)(82,89)(83,88)(84,87)(85,86)(91,100)(92,99)(93,98)(94,97)(95,96)(121,140)(122,139)(123,138)(124,137)(125,136)(126,135)(127,134)(128,133)(129,132)(130,131)>;
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), (1,98,27,47,146,62,111,133)(2,81,36,54,147,65,120,140)(3,84,25,41,148,68,109,127)(4,87,34,48,149,71,118,134)(5,90,23,55,150,74,107,121)(6,93,32,42,151,77,116,128)(7,96,21,49,152,80,105,135)(8,99,30,56,153,63,114,122)(9,82,39,43,154,66,103,129)(10,85,28,50,155,69,112,136)(11,88,37,57,156,72,101,123)(12,91,26,44,157,75,110,130)(13,94,35,51,158,78,119,137)(14,97,24,58,159,61,108,124)(15,100,33,45,160,64,117,131)(16,83,22,52,141,67,106,138)(17,86,31,59,142,70,115,125)(18,89,40,46,143,73,104,132)(19,92,29,53,144,76,113,139)(20,95,38,60,145,79,102,126), (1,151)(2,150)(3,149)(4,148)(5,147)(6,146)(7,145)(8,144)(9,143)(10,142)(11,141)(12,160)(13,159)(14,158)(15,157)(16,156)(17,155)(18,154)(19,153)(20,152)(21,102)(22,101)(23,120)(24,119)(25,118)(26,117)(27,116)(28,115)(29,114)(30,113)(31,112)(32,111)(33,110)(34,109)(35,108)(36,107)(37,106)(38,105)(39,104)(40,103)(41,48)(42,47)(43,46)(44,45)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)(79,80)(81,90)(82,89)(83,88)(84,87)(85,86)(91,100)(92,99)(93,98)(94,97)(95,96)(121,140)(122,139)(123,138)(124,137)(125,136)(126,135)(127,134)(128,133)(129,132)(130,131) );
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)], [(1,98,27,47,146,62,111,133),(2,81,36,54,147,65,120,140),(3,84,25,41,148,68,109,127),(4,87,34,48,149,71,118,134),(5,90,23,55,150,74,107,121),(6,93,32,42,151,77,116,128),(7,96,21,49,152,80,105,135),(8,99,30,56,153,63,114,122),(9,82,39,43,154,66,103,129),(10,85,28,50,155,69,112,136),(11,88,37,57,156,72,101,123),(12,91,26,44,157,75,110,130),(13,94,35,51,158,78,119,137),(14,97,24,58,159,61,108,124),(15,100,33,45,160,64,117,131),(16,83,22,52,141,67,106,138),(17,86,31,59,142,70,115,125),(18,89,40,46,143,73,104,132),(19,92,29,53,144,76,113,139),(20,95,38,60,145,79,102,126)], [(1,151),(2,150),(3,149),(4,148),(5,147),(6,146),(7,145),(8,144),(9,143),(10,142),(11,141),(12,160),(13,159),(14,158),(15,157),(16,156),(17,155),(18,154),(19,153),(20,152),(21,102),(22,101),(23,120),(24,119),(25,118),(26,117),(27,116),(28,115),(29,114),(30,113),(31,112),(32,111),(33,110),(34,109),(35,108),(36,107),(37,106),(38,105),(39,104),(40,103),(41,48),(42,47),(43,46),(44,45),(49,60),(50,59),(51,58),(52,57),(53,56),(54,55),(61,78),(62,77),(63,76),(64,75),(65,74),(66,73),(67,72),(68,71),(69,70),(79,80),(81,90),(82,89),(83,88),(84,87),(85,86),(91,100),(92,99),(93,98),(94,97),(95,96),(121,140),(122,139),(123,138),(124,137),(125,136),(126,135),(127,134),(128,133),(129,132),(130,131)]])
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 | 20A | ··· | 20F |
| 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 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 2 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 4 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 4 | 4 | 4 | 8 | ··· | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | C4○D4 | M4(2) | C8○D4 | F5 | C2×F5 | C4.F5 | D4×F5 | Q8.F5 |
| kernel | C20⋊M4(2) | C4×C5⋊C8 | C20⋊C8 | D10⋊C8 | D20⋊8C4 | C2×C4.F5 | D10⋊C4 | C5×C4⋊C4 | C2×D20 | C5⋊C8 | Dic5 | C20 | C10 | C4⋊C4 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 3 | 4 | 1 | 1 |
Matrix representation of C20⋊M4(2) ►in GL6(𝔽41)
| 1 | 2 | 0 | 0 | 0 | 0 |
| 40 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 | 0 | 1 |
| 0 | 0 | 0 | 40 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 1 |
| 40 | 39 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 8 | 2 | 2 |
| 0 | 0 | 23 | 16 | 23 | 10 |
| 0 | 0 | 31 | 18 | 25 | 18 |
| 0 | 0 | 39 | 39 | 33 | 20 |
| 40 | 39 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
G:=sub<GL(6,GF(41))| [1,40,0,0,0,0,2,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,1,1,1],[40,0,0,0,0,0,39,1,0,0,0,0,0,0,21,23,31,39,0,0,8,16,18,39,0,0,2,23,25,33,0,0,2,10,18,20],[40,0,0,0,0,0,39,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0] >;
C20⋊M4(2) in GAP, Magma, Sage, TeX
C_{20}\rtimes M_4(2) % in TeX
G:=Group("C20:M4(2)"); // GroupNames label
G:=SmallGroup(320,1043);
// by ID
G=gap.SmallGroup(320,1043);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,758,219,184,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^7,c*a*c=a^-1,c*b*c=b^5>;
// generators/relations