metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊8C4⋊12C2, C22⋊C8.8D5, (C8×Dic5)⋊14C2, (C2×C8).193D10, C4⋊Dic5.22C4, C23.11(C4×D5), C10.31(C8○D4), C20.8Q8⋊17C2, (C22×C4).74D10, C23.D5.10C4, C20.296(C4○D4), (C2×C40).169C22, (C2×C20).818C23, C20.55D4.2C2, C4.122(D4⋊2D5), C2.9(D20.2C4), C2.9(D20.3C4), (C22×C20).91C22, C5⋊5(C42.7C22), C10.41(C42⋊C2), (C4×Dic5).301C22, C23.21D10.2C2, C2.10(C23.11D10), (C2×C4).30(C4×D5), C22.102(C2×C4×D5), (C2×C20).211(C2×C4), (C5×C22⋊C8).11C2, (C2×Dic5).18(C2×C4), (C2×C4).760(C22×D5), (C2×C10).174(C22×C4), (C22×C10).104(C2×C4), (C2×C5⋊2C8).307C22, SmallGroup(320,347)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊8C4⋊C2
G = < a,b,c | a40=b4=c2=1, bab-1=a29, cac=ab2, cbc=a20b >
Subgroups: 254 in 96 conjugacy classes, 47 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, Dic5, C20, C20, C2×C10, C2×C10, C4×C8, C8⋊C4, C22⋊C8, C22⋊C8, C4⋊C8, C42⋊C2, C5⋊2C8, C40, C2×Dic5, C2×C20, C2×C20, C22×C10, C42.7C22, C2×C5⋊2C8, C4×Dic5, C4⋊Dic5, C23.D5, C2×C40, C22×C20, C8×Dic5, C20.8Q8, C40⋊8C4, C20.55D4, C5×C22⋊C8, C23.21D10, C40⋊8C4⋊C2
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C4○D4, D10, C42⋊C2, C8○D4, C4×D5, C22×D5, C42.7C22, C2×C4×D5, D4⋊2D5, C23.11D10, D20.3C4, D20.2C4, C40⋊8C4⋊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 77 160 106)(2 66 121 95)(3 55 122 84)(4 44 123 113)(5 73 124 102)(6 62 125 91)(7 51 126 120)(8 80 127 109)(9 69 128 98)(10 58 129 87)(11 47 130 116)(12 76 131 105)(13 65 132 94)(14 54 133 83)(15 43 134 112)(16 72 135 101)(17 61 136 90)(18 50 137 119)(19 79 138 108)(20 68 139 97)(21 57 140 86)(22 46 141 115)(23 75 142 104)(24 64 143 93)(25 53 144 82)(26 42 145 111)(27 71 146 100)(28 60 147 89)(29 49 148 118)(30 78 149 107)(31 67 150 96)(32 56 151 85)(33 45 152 114)(34 74 153 103)(35 63 154 92)(36 52 155 81)(37 41 156 110)(38 70 157 99)(39 59 158 88)(40 48 159 117)
(2 121)(4 123)(6 125)(8 127)(10 129)(12 131)(14 133)(16 135)(18 137)(20 139)(22 141)(24 143)(26 145)(28 147)(30 149)(32 151)(34 153)(36 155)(38 157)(40 159)(41 61)(42 91)(43 63)(44 93)(45 65)(46 95)(47 67)(48 97)(49 69)(50 99)(51 71)(52 101)(53 73)(54 103)(55 75)(56 105)(57 77)(58 107)(59 79)(60 109)(62 111)(64 113)(66 115)(68 117)(70 119)(72 81)(74 83)(76 85)(78 87)(80 89)(82 102)(84 104)(86 106)(88 108)(90 110)(92 112)(94 114)(96 116)(98 118)(100 120)
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), (1,77,160,106)(2,66,121,95)(3,55,122,84)(4,44,123,113)(5,73,124,102)(6,62,125,91)(7,51,126,120)(8,80,127,109)(9,69,128,98)(10,58,129,87)(11,47,130,116)(12,76,131,105)(13,65,132,94)(14,54,133,83)(15,43,134,112)(16,72,135,101)(17,61,136,90)(18,50,137,119)(19,79,138,108)(20,68,139,97)(21,57,140,86)(22,46,141,115)(23,75,142,104)(24,64,143,93)(25,53,144,82)(26,42,145,111)(27,71,146,100)(28,60,147,89)(29,49,148,118)(30,78,149,107)(31,67,150,96)(32,56,151,85)(33,45,152,114)(34,74,153,103)(35,63,154,92)(36,52,155,81)(37,41,156,110)(38,70,157,99)(39,59,158,88)(40,48,159,117), (2,121)(4,123)(6,125)(8,127)(10,129)(12,131)(14,133)(16,135)(18,137)(20,139)(22,141)(24,143)(26,145)(28,147)(30,149)(32,151)(34,153)(36,155)(38,157)(40,159)(41,61)(42,91)(43,63)(44,93)(45,65)(46,95)(47,67)(48,97)(49,69)(50,99)(51,71)(52,101)(53,73)(54,103)(55,75)(56,105)(57,77)(58,107)(59,79)(60,109)(62,111)(64,113)(66,115)(68,117)(70,119)(72,81)(74,83)(76,85)(78,87)(80,89)(82,102)(84,104)(86,106)(88,108)(90,110)(92,112)(94,114)(96,116)(98,118)(100,120)>;
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), (1,77,160,106)(2,66,121,95)(3,55,122,84)(4,44,123,113)(5,73,124,102)(6,62,125,91)(7,51,126,120)(8,80,127,109)(9,69,128,98)(10,58,129,87)(11,47,130,116)(12,76,131,105)(13,65,132,94)(14,54,133,83)(15,43,134,112)(16,72,135,101)(17,61,136,90)(18,50,137,119)(19,79,138,108)(20,68,139,97)(21,57,140,86)(22,46,141,115)(23,75,142,104)(24,64,143,93)(25,53,144,82)(26,42,145,111)(27,71,146,100)(28,60,147,89)(29,49,148,118)(30,78,149,107)(31,67,150,96)(32,56,151,85)(33,45,152,114)(34,74,153,103)(35,63,154,92)(36,52,155,81)(37,41,156,110)(38,70,157,99)(39,59,158,88)(40,48,159,117), (2,121)(4,123)(6,125)(8,127)(10,129)(12,131)(14,133)(16,135)(18,137)(20,139)(22,141)(24,143)(26,145)(28,147)(30,149)(32,151)(34,153)(36,155)(38,157)(40,159)(41,61)(42,91)(43,63)(44,93)(45,65)(46,95)(47,67)(48,97)(49,69)(50,99)(51,71)(52,101)(53,73)(54,103)(55,75)(56,105)(57,77)(58,107)(59,79)(60,109)(62,111)(64,113)(66,115)(68,117)(70,119)(72,81)(74,83)(76,85)(78,87)(80,89)(82,102)(84,104)(86,106)(88,108)(90,110)(92,112)(94,114)(96,116)(98,118)(100,120) );
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)], [(1,77,160,106),(2,66,121,95),(3,55,122,84),(4,44,123,113),(5,73,124,102),(6,62,125,91),(7,51,126,120),(8,80,127,109),(9,69,128,98),(10,58,129,87),(11,47,130,116),(12,76,131,105),(13,65,132,94),(14,54,133,83),(15,43,134,112),(16,72,135,101),(17,61,136,90),(18,50,137,119),(19,79,138,108),(20,68,139,97),(21,57,140,86),(22,46,141,115),(23,75,142,104),(24,64,143,93),(25,53,144,82),(26,42,145,111),(27,71,146,100),(28,60,147,89),(29,49,148,118),(30,78,149,107),(31,67,150,96),(32,56,151,85),(33,45,152,114),(34,74,153,103),(35,63,154,92),(36,52,155,81),(37,41,156,110),(38,70,157,99),(39,59,158,88),(40,48,159,117)], [(2,121),(4,123),(6,125),(8,127),(10,129),(12,131),(14,133),(16,135),(18,137),(20,139),(22,141),(24,143),(26,145),(28,147),(30,149),(32,151),(34,153),(36,155),(38,157),(40,159),(41,61),(42,91),(43,63),(44,93),(45,65),(46,95),(47,67),(48,97),(49,69),(50,99),(51,71),(52,101),(53,73),(54,103),(55,75),(56,105),(57,77),(58,107),(59,79),(60,109),(62,111),(64,113),(66,115),(68,117),(70,119),(72,81),(74,83),(76,85),(78,87),(80,89),(82,102),(84,104),(86,106),(88,108),(90,110),(92,112),(94,114),(96,116),(98,118),(100,120)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D5 | C4○D4 | D10 | D10 | C8○D4 | C4×D5 | C4×D5 | D20.3C4 | D4⋊2D5 | D20.2C4 |
| kernel | C40⋊8C4⋊C2 | C8×Dic5 | C20.8Q8 | C40⋊8C4 | C20.55D4 | C5×C22⋊C8 | C23.21D10 | C4⋊Dic5 | C23.D5 | C22⋊C8 | C20 | C2×C8 | C22×C4 | C10 | C2×C4 | C23 | C2 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 4 | 4 | 2 | 8 | 4 | 4 | 16 | 4 | 4 |
Matrix representation of C40⋊8C4⋊C2 ►in GL4(𝔽41) generated by
| 17 | 0 | 0 | 0 |
| 24 | 15 | 0 | 0 |
| 0 | 0 | 9 | 20 |
| 0 | 0 | 37 | 32 |
| 1 | 17 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 |
| 24 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 36 | 40 |
G:=sub<GL(4,GF(41))| [17,24,0,0,0,15,0,0,0,0,9,37,0,0,20,32],[1,0,0,0,17,40,0,0,0,0,9,0,0,0,0,9],[1,24,0,0,0,40,0,0,0,0,1,36,0,0,0,40] >;
C40⋊8C4⋊C2 in GAP, Magma, Sage, TeX
C_{40}\rtimes_8C_4\rtimes C_2 % in TeX
G:=Group("C40:8C4:C2"); // GroupNames label
G:=SmallGroup(320,347);
// by ID
G=gap.SmallGroup(320,347);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,477,219,58,136,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^4=c^2=1,b*a*b^-1=a^29,c*a*c=a*b^2,c*b*c=a^20*b>;
// generators/relations