metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊5C4⋊9C2, D4⋊C4⋊7D5, (C2×C8).11D10, D10⋊1C8⋊7C2, C4⋊C4.138D10, (C2×D4).30D10, C20⋊2D4.7C2, C20.Q8⋊7C2, C10.25(C4○D8), C4.54(C4○D20), D4⋊Dic5⋊10C2, (C2×C40).11C22, (C22×D5).24D4, C22.182(D4×D5), C20.152(C4○D4), C2.11(D8⋊3D5), C4.81(D4⋊2D5), C2.12(D40⋊C2), C10.56(C8⋊C22), (C2×C20).224C23, (C2×Dic5).200D4, (D4×C10).45C22, C5⋊3(C23.19D4), C4⋊Dic5.76C22, C2.15(D10.12D4), C10.23(C22.D4), C4⋊C4⋊7D5⋊4C2, (C5×D4⋊C4)⋊7C2, (C2×C4×D5).20C22, (C2×C10).237(C2×D4), (C5×C4⋊C4).25C22, (C2×C5⋊2C8).22C22, (C2×C4).331(C22×D5), SmallGroup(320,411)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊5C4⋊C2
G = < a,b,c | a40=b4=c2=1, bab-1=a-1, cac=a11b2, cbc=b-1 >
Subgroups: 446 in 106 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C2×D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, D4⋊C4, C4.Q8, C2.D8, C42⋊C2, C4⋊D4, C5⋊2C8, C40, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C23.19D4, C2×C5⋊2C8, C4×Dic5, C4⋊Dic5, D10⋊C4, C23.D5, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×C5⋊D4, D4×C10, C20.Q8, C40⋊5C4, D10⋊1C8, D4⋊Dic5, C5×D4⋊C4, C4⋊C4⋊7D5, C20⋊2D4, C40⋊5C4⋊C2
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C22.D4, C4○D8, C8⋊C22, C22×D5, C23.19D4, C4○D20, D4×D5, D4⋊2D5, D10.12D4, D8⋊3D5, D40⋊C2, C40⋊5C4⋊C2
►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)
(1 133 46 113)(2 132 47 112)(3 131 48 111)(4 130 49 110)(5 129 50 109)(6 128 51 108)(7 127 52 107)(8 126 53 106)(9 125 54 105)(10 124 55 104)(11 123 56 103)(12 122 57 102)(13 121 58 101)(14 160 59 100)(15 159 60 99)(16 158 61 98)(17 157 62 97)(18 156 63 96)(19 155 64 95)(20 154 65 94)(21 153 66 93)(22 152 67 92)(23 151 68 91)(24 150 69 90)(25 149 70 89)(26 148 71 88)(27 147 72 87)(28 146 73 86)(29 145 74 85)(30 144 75 84)(31 143 76 83)(32 142 77 82)(33 141 78 81)(34 140 79 120)(35 139 80 119)(36 138 41 118)(37 137 42 117)(38 136 43 116)(39 135 44 115)(40 134 45 114)
(1 133)(2 84)(3 155)(4 106)(5 137)(6 88)(7 159)(8 110)(9 141)(10 92)(11 123)(12 114)(13 145)(14 96)(15 127)(16 118)(17 149)(18 100)(19 131)(20 82)(21 153)(22 104)(23 135)(24 86)(25 157)(26 108)(27 139)(28 90)(29 121)(30 112)(31 143)(32 94)(33 125)(34 116)(35 147)(36 98)(37 129)(38 120)(39 151)(40 102)(41 158)(42 109)(43 140)(44 91)(45 122)(46 113)(47 144)(48 95)(49 126)(50 117)(51 148)(52 99)(53 130)(54 81)(55 152)(56 103)(57 134)(58 85)(59 156)(60 107)(61 138)(62 89)(63 160)(64 111)(65 142)(66 93)(67 124)(68 115)(69 146)(70 97)(71 128)(72 119)(73 150)(74 101)(75 132)(76 83)(77 154)(78 105)(79 136)(80 87)
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), (1,133,46,113)(2,132,47,112)(3,131,48,111)(4,130,49,110)(5,129,50,109)(6,128,51,108)(7,127,52,107)(8,126,53,106)(9,125,54,105)(10,124,55,104)(11,123,56,103)(12,122,57,102)(13,121,58,101)(14,160,59,100)(15,159,60,99)(16,158,61,98)(17,157,62,97)(18,156,63,96)(19,155,64,95)(20,154,65,94)(21,153,66,93)(22,152,67,92)(23,151,68,91)(24,150,69,90)(25,149,70,89)(26,148,71,88)(27,147,72,87)(28,146,73,86)(29,145,74,85)(30,144,75,84)(31,143,76,83)(32,142,77,82)(33,141,78,81)(34,140,79,120)(35,139,80,119)(36,138,41,118)(37,137,42,117)(38,136,43,116)(39,135,44,115)(40,134,45,114), (1,133)(2,84)(3,155)(4,106)(5,137)(6,88)(7,159)(8,110)(9,141)(10,92)(11,123)(12,114)(13,145)(14,96)(15,127)(16,118)(17,149)(18,100)(19,131)(20,82)(21,153)(22,104)(23,135)(24,86)(25,157)(26,108)(27,139)(28,90)(29,121)(30,112)(31,143)(32,94)(33,125)(34,116)(35,147)(36,98)(37,129)(38,120)(39,151)(40,102)(41,158)(42,109)(43,140)(44,91)(45,122)(46,113)(47,144)(48,95)(49,126)(50,117)(51,148)(52,99)(53,130)(54,81)(55,152)(56,103)(57,134)(58,85)(59,156)(60,107)(61,138)(62,89)(63,160)(64,111)(65,142)(66,93)(67,124)(68,115)(69,146)(70,97)(71,128)(72,119)(73,150)(74,101)(75,132)(76,83)(77,154)(78,105)(79,136)(80,87)>;
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), (1,133,46,113)(2,132,47,112)(3,131,48,111)(4,130,49,110)(5,129,50,109)(6,128,51,108)(7,127,52,107)(8,126,53,106)(9,125,54,105)(10,124,55,104)(11,123,56,103)(12,122,57,102)(13,121,58,101)(14,160,59,100)(15,159,60,99)(16,158,61,98)(17,157,62,97)(18,156,63,96)(19,155,64,95)(20,154,65,94)(21,153,66,93)(22,152,67,92)(23,151,68,91)(24,150,69,90)(25,149,70,89)(26,148,71,88)(27,147,72,87)(28,146,73,86)(29,145,74,85)(30,144,75,84)(31,143,76,83)(32,142,77,82)(33,141,78,81)(34,140,79,120)(35,139,80,119)(36,138,41,118)(37,137,42,117)(38,136,43,116)(39,135,44,115)(40,134,45,114), (1,133)(2,84)(3,155)(4,106)(5,137)(6,88)(7,159)(8,110)(9,141)(10,92)(11,123)(12,114)(13,145)(14,96)(15,127)(16,118)(17,149)(18,100)(19,131)(20,82)(21,153)(22,104)(23,135)(24,86)(25,157)(26,108)(27,139)(28,90)(29,121)(30,112)(31,143)(32,94)(33,125)(34,116)(35,147)(36,98)(37,129)(38,120)(39,151)(40,102)(41,158)(42,109)(43,140)(44,91)(45,122)(46,113)(47,144)(48,95)(49,126)(50,117)(51,148)(52,99)(53,130)(54,81)(55,152)(56,103)(57,134)(58,85)(59,156)(60,107)(61,138)(62,89)(63,160)(64,111)(65,142)(66,93)(67,124)(68,115)(69,146)(70,97)(71,128)(72,119)(73,150)(74,101)(75,132)(76,83)(77,154)(78,105)(79,136)(80,87) );
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)], [(1,133,46,113),(2,132,47,112),(3,131,48,111),(4,130,49,110),(5,129,50,109),(6,128,51,108),(7,127,52,107),(8,126,53,106),(9,125,54,105),(10,124,55,104),(11,123,56,103),(12,122,57,102),(13,121,58,101),(14,160,59,100),(15,159,60,99),(16,158,61,98),(17,157,62,97),(18,156,63,96),(19,155,64,95),(20,154,65,94),(21,153,66,93),(22,152,67,92),(23,151,68,91),(24,150,69,90),(25,149,70,89),(26,148,71,88),(27,147,72,87),(28,146,73,86),(29,145,74,85),(30,144,75,84),(31,143,76,83),(32,142,77,82),(33,141,78,81),(34,140,79,120),(35,139,80,119),(36,138,41,118),(37,137,42,117),(38,136,43,116),(39,135,44,115),(40,134,45,114)], [(1,133),(2,84),(3,155),(4,106),(5,137),(6,88),(7,159),(8,110),(9,141),(10,92),(11,123),(12,114),(13,145),(14,96),(15,127),(16,118),(17,149),(18,100),(19,131),(20,82),(21,153),(22,104),(23,135),(24,86),(25,157),(26,108),(27,139),(28,90),(29,121),(30,112),(31,143),(32,94),(33,125),(34,116),(35,147),(36,98),(37,129),(38,120),(39,151),(40,102),(41,158),(42,109),(43,140),(44,91),(45,122),(46,113),(47,144),(48,95),(49,126),(50,117),(51,148),(52,99),(53,130),(54,81),(55,152),(56,103),(57,134),(58,85),(59,156),(60,107),(61,138),(62,89),(63,160),(64,111),(65,142),(66,93),(67,124),(68,115),(69,146),(70,97),(71,128),(72,119),(73,150),(74,101),(75,132),(76,83),(77,154),(78,105),(79,136),(80,87)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 8 | 20 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C4○D8 | C4○D20 | C8⋊C22 | D4⋊2D5 | D4×D5 | D8⋊3D5 | D40⋊C2 |
| kernel | C40⋊5C4⋊C2 | C20.Q8 | C40⋊5C4 | D10⋊1C8 | D4⋊Dic5 | C5×D4⋊C4 | C4⋊C4⋊7D5 | C20⋊2D4 | C2×Dic5 | C22×D5 | D4⋊C4 | C20 | C4⋊C4 | C2×C8 | C2×D4 | C10 | C4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 4 | 8 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of C40⋊5C4⋊C2 ►in GL4(𝔽41) generated by
| 3 | 0 | 0 | 0 |
| 8 | 14 | 0 | 0 |
| 0 | 0 | 39 | 16 |
| 0 | 0 | 25 | 25 |
| 29 | 4 | 0 | 0 |
| 15 | 12 | 0 | 0 |
| 0 | 0 | 20 | 20 |
| 0 | 0 | 23 | 21 |
| 12 | 37 | 0 | 0 |
| 5 | 29 | 0 | 0 |
| 0 | 0 | 18 | 6 |
| 0 | 0 | 35 | 23 |
G:=sub<GL(4,GF(41))| [3,8,0,0,0,14,0,0,0,0,39,25,0,0,16,25],[29,15,0,0,4,12,0,0,0,0,20,23,0,0,20,21],[12,5,0,0,37,29,0,0,0,0,18,35,0,0,6,23] >;
C40⋊5C4⋊C2 in GAP, Magma, Sage, TeX
C_{40}\rtimes_5C_4\rtimes C_2 % in TeX
G:=Group("C40:5C4:C2"); // GroupNames label
G:=SmallGroup(320,411);
// by ID
G=gap.SmallGroup(320,411);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,64,926,219,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^11*b^2,c*b*c=b^-1>;
// generators/relations