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 [×3], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×2], C2×C4, C2×C4 [×6], D4 [×4], C23 [×2], D5, C10 [×3], C10, C42, C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C2×C8, C2×C8, C22×C4, C2×D4, C2×D4, Dic5 [×3], C20 [×2], C20, D10 [×3], C2×C10, C2×C10 [×3], C22⋊C8, D4⋊C4, D4⋊C4, C4.Q8, C2.D8, C42⋊C2, C4⋊D4, C5⋊2C8, C40, C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×2], C2×C20, C2×C20, C5×D4 [×2], C22×D5, C22×C10, C23.19D4, C2×C5⋊2C8, C4×Dic5, C4⋊Dic5 [×2], 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 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], 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 51 122 117)(2 50 123 116)(3 49 124 115)(4 48 125 114)(5 47 126 113)(6 46 127 112)(7 45 128 111)(8 44 129 110)(9 43 130 109)(10 42 131 108)(11 41 132 107)(12 80 133 106)(13 79 134 105)(14 78 135 104)(15 77 136 103)(16 76 137 102)(17 75 138 101)(18 74 139 100)(19 73 140 99)(20 72 141 98)(21 71 142 97)(22 70 143 96)(23 69 144 95)(24 68 145 94)(25 67 146 93)(26 66 147 92)(27 65 148 91)(28 64 149 90)(29 63 150 89)(30 62 151 88)(31 61 152 87)(32 60 153 86)(33 59 154 85)(34 58 155 84)(35 57 156 83)(36 56 157 82)(37 55 158 81)(38 54 159 120)(39 53 160 119)(40 52 121 118)
(1 51)(2 88)(3 73)(4 110)(5 55)(6 92)(7 77)(8 114)(9 59)(10 96)(11 41)(12 118)(13 63)(14 100)(15 45)(16 82)(17 67)(18 104)(19 49)(20 86)(21 71)(22 108)(23 53)(24 90)(25 75)(26 112)(27 57)(28 94)(29 79)(30 116)(31 61)(32 98)(33 43)(34 120)(35 65)(36 102)(37 47)(38 84)(39 69)(40 106)(42 143)(44 125)(46 147)(48 129)(50 151)(52 133)(54 155)(56 137)(58 159)(60 141)(62 123)(64 145)(66 127)(68 149)(70 131)(72 153)(74 135)(76 157)(78 139)(80 121)(81 126)(83 148)(85 130)(87 152)(89 134)(91 156)(93 138)(95 160)(97 142)(99 124)(101 146)(103 128)(105 150)(107 132)(109 154)(111 136)(113 158)(115 140)(117 122)(119 144)
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), (1,51,122,117)(2,50,123,116)(3,49,124,115)(4,48,125,114)(5,47,126,113)(6,46,127,112)(7,45,128,111)(8,44,129,110)(9,43,130,109)(10,42,131,108)(11,41,132,107)(12,80,133,106)(13,79,134,105)(14,78,135,104)(15,77,136,103)(16,76,137,102)(17,75,138,101)(18,74,139,100)(19,73,140,99)(20,72,141,98)(21,71,142,97)(22,70,143,96)(23,69,144,95)(24,68,145,94)(25,67,146,93)(26,66,147,92)(27,65,148,91)(28,64,149,90)(29,63,150,89)(30,62,151,88)(31,61,152,87)(32,60,153,86)(33,59,154,85)(34,58,155,84)(35,57,156,83)(36,56,157,82)(37,55,158,81)(38,54,159,120)(39,53,160,119)(40,52,121,118), (1,51)(2,88)(3,73)(4,110)(5,55)(6,92)(7,77)(8,114)(9,59)(10,96)(11,41)(12,118)(13,63)(14,100)(15,45)(16,82)(17,67)(18,104)(19,49)(20,86)(21,71)(22,108)(23,53)(24,90)(25,75)(26,112)(27,57)(28,94)(29,79)(30,116)(31,61)(32,98)(33,43)(34,120)(35,65)(36,102)(37,47)(38,84)(39,69)(40,106)(42,143)(44,125)(46,147)(48,129)(50,151)(52,133)(54,155)(56,137)(58,159)(60,141)(62,123)(64,145)(66,127)(68,149)(70,131)(72,153)(74,135)(76,157)(78,139)(80,121)(81,126)(83,148)(85,130)(87,152)(89,134)(91,156)(93,138)(95,160)(97,142)(99,124)(101,146)(103,128)(105,150)(107,132)(109,154)(111,136)(113,158)(115,140)(117,122)(119,144)>;
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), (1,51,122,117)(2,50,123,116)(3,49,124,115)(4,48,125,114)(5,47,126,113)(6,46,127,112)(7,45,128,111)(8,44,129,110)(9,43,130,109)(10,42,131,108)(11,41,132,107)(12,80,133,106)(13,79,134,105)(14,78,135,104)(15,77,136,103)(16,76,137,102)(17,75,138,101)(18,74,139,100)(19,73,140,99)(20,72,141,98)(21,71,142,97)(22,70,143,96)(23,69,144,95)(24,68,145,94)(25,67,146,93)(26,66,147,92)(27,65,148,91)(28,64,149,90)(29,63,150,89)(30,62,151,88)(31,61,152,87)(32,60,153,86)(33,59,154,85)(34,58,155,84)(35,57,156,83)(36,56,157,82)(37,55,158,81)(38,54,159,120)(39,53,160,119)(40,52,121,118), (1,51)(2,88)(3,73)(4,110)(5,55)(6,92)(7,77)(8,114)(9,59)(10,96)(11,41)(12,118)(13,63)(14,100)(15,45)(16,82)(17,67)(18,104)(19,49)(20,86)(21,71)(22,108)(23,53)(24,90)(25,75)(26,112)(27,57)(28,94)(29,79)(30,116)(31,61)(32,98)(33,43)(34,120)(35,65)(36,102)(37,47)(38,84)(39,69)(40,106)(42,143)(44,125)(46,147)(48,129)(50,151)(52,133)(54,155)(56,137)(58,159)(60,141)(62,123)(64,145)(66,127)(68,149)(70,131)(72,153)(74,135)(76,157)(78,139)(80,121)(81,126)(83,148)(85,130)(87,152)(89,134)(91,156)(93,138)(95,160)(97,142)(99,124)(101,146)(103,128)(105,150)(107,132)(109,154)(111,136)(113,158)(115,140)(117,122)(119,144) );
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)], [(1,51,122,117),(2,50,123,116),(3,49,124,115),(4,48,125,114),(5,47,126,113),(6,46,127,112),(7,45,128,111),(8,44,129,110),(9,43,130,109),(10,42,131,108),(11,41,132,107),(12,80,133,106),(13,79,134,105),(14,78,135,104),(15,77,136,103),(16,76,137,102),(17,75,138,101),(18,74,139,100),(19,73,140,99),(20,72,141,98),(21,71,142,97),(22,70,143,96),(23,69,144,95),(24,68,145,94),(25,67,146,93),(26,66,147,92),(27,65,148,91),(28,64,149,90),(29,63,150,89),(30,62,151,88),(31,61,152,87),(32,60,153,86),(33,59,154,85),(34,58,155,84),(35,57,156,83),(36,56,157,82),(37,55,158,81),(38,54,159,120),(39,53,160,119),(40,52,121,118)], [(1,51),(2,88),(3,73),(4,110),(5,55),(6,92),(7,77),(8,114),(9,59),(10,96),(11,41),(12,118),(13,63),(14,100),(15,45),(16,82),(17,67),(18,104),(19,49),(20,86),(21,71),(22,108),(23,53),(24,90),(25,75),(26,112),(27,57),(28,94),(29,79),(30,116),(31,61),(32,98),(33,43),(34,120),(35,65),(36,102),(37,47),(38,84),(39,69),(40,106),(42,143),(44,125),(46,147),(48,129),(50,151),(52,133),(54,155),(56,137),(58,159),(60,141),(62,123),(64,145),(66,127),(68,149),(70,131),(72,153),(74,135),(76,157),(78,139),(80,121),(81,126),(83,148),(85,130),(87,152),(89,134),(91,156),(93,138),(95,160),(97,142),(99,124),(101,146),(103,128),(105,150),(107,132),(109,154),(111,136),(113,158),(115,140),(117,122),(119,144)])
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