direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C40.C4, (C8×D5).5C4, (C2×C8).15F5, C8.28(C2×F5), C40.29(C2×C4), (C2×C40).15C4, (C4×D5).84D4, C4.14(C4⋊F5), C20.21(C4⋊C4), (C4×D5).26Q8, D10.12(C2×Q8), C10⋊1(C8.C4), D10.30(C4⋊C4), C4.38(C22×F5), C4.F5.7C22, C20.78(C22×C4), Dic5.31(C2×D4), (C4×D5).78C23, (C8×D5).57C22, C22.24(C4⋊F5), (C22×D5).18Q8, Dic5.31(C4⋊C4), (C2×Dic5).175D4, C5⋊1(C2×C8.C4), (D5×C2×C8).25C2, C2.17(C2×C4⋊F5), C10.14(C2×C4⋊C4), (C2×C5⋊2C8).24C4, (C2×C4.F5).9C2, C5⋊2C8.48(C2×C4), (C4×D5).87(C2×C4), (C2×C4).139(C2×F5), (C2×C10).22(C4⋊C4), (C2×C20).128(C2×C4), (C2×C4×D5).396C22, SmallGroup(320,1060)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 346 in 106 conjugacy classes, 52 normal (28 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C22, C22 [×4], C5, C8 [×2], C8 [×6], C2×C4, C2×C4 [×5], C23, D5 [×2], C10, C10 [×2], C2×C8, C2×C8 [×7], M4(2) [×6], C22×C4, Dic5 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C8.C4 [×4], C22×C8, C2×M4(2) [×2], C5⋊2C8 [×2], C40 [×2], C5⋊C8 [×4], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, C2×C8.C4, C8×D5 [×4], C2×C5⋊2C8, C2×C40, C4.F5 [×4], C4.F5 [×2], C2×C5⋊C8 [×2], C2×C4×D5, C40.C4 [×4], D5×C2×C8, C2×C4.F5 [×2], C2×C40.C4
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, F5, C8.C4 [×2], C2×C4⋊C4, C2×F5 [×3], C2×C8.C4, C4⋊F5 [×2], C22×F5, C40.C4 [×2], C2×C4⋊F5, C2×C40.C4
Generators and relations
G = < a,b,c | a2=b40=1, c4=b20, ab=ba, ac=ca, cbc-1=b3 >
►On 160 points
Generators in S160
(1 120)(2 81)(3 82)(4 83)(5 84)(6 85)(7 86)(8 87)(9 88)(10 89)(11 90)(12 91)(13 92)(14 93)(15 94)(16 95)(17 96)(18 97)(19 98)(20 99)(21 100)(22 101)(23 102)(24 103)(25 104)(26 105)(27 106)(28 107)(29 108)(30 109)(31 110)(32 111)(33 112)(34 113)(35 114)(36 115)(37 116)(38 117)(39 118)(40 119)(41 146)(42 147)(43 148)(44 149)(45 150)(46 151)(47 152)(48 153)(49 154)(50 155)(51 156)(52 157)(53 158)(54 159)(55 160)(56 121)(57 122)(58 123)(59 124)(60 125)(61 126)(62 127)(63 128)(64 129)(65 130)(66 131)(67 132)(68 133)(69 134)(70 135)(71 136)(72 137)(73 138)(74 139)(75 140)(76 141)(77 142)(78 143)(79 144)(80 145)
(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 63 110 138 21 43 90 158)(2 50 119 141 22 70 99 121)(3 77 88 144 23 57 108 124)(4 64 97 147 24 44 117 127)(5 51 106 150 25 71 86 130)(6 78 115 153 26 58 95 133)(7 65 84 156 27 45 104 136)(8 52 93 159 28 72 113 139)(9 79 102 122 29 59 82 142)(10 66 111 125 30 46 91 145)(11 53 120 128 31 73 100 148)(12 80 89 131 32 60 109 151)(13 67 98 134 33 47 118 154)(14 54 107 137 34 74 87 157)(15 41 116 140 35 61 96 160)(16 68 85 143 36 48 105 123)(17 55 94 146 37 75 114 126)(18 42 103 149 38 62 83 129)(19 69 112 152 39 49 92 132)(20 56 81 155 40 76 101 135)
G:=sub<Sym(160)| (1,120)(2,81)(3,82)(4,83)(5,84)(6,85)(7,86)(8,87)(9,88)(10,89)(11,90)(12,91)(13,92)(14,93)(15,94)(16,95)(17,96)(18,97)(19,98)(20,99)(21,100)(22,101)(23,102)(24,103)(25,104)(26,105)(27,106)(28,107)(29,108)(30,109)(31,110)(32,111)(33,112)(34,113)(35,114)(36,115)(37,116)(38,117)(39,118)(40,119)(41,146)(42,147)(43,148)(44,149)(45,150)(46,151)(47,152)(48,153)(49,154)(50,155)(51,156)(52,157)(53,158)(54,159)(55,160)(56,121)(57,122)(58,123)(59,124)(60,125)(61,126)(62,127)(63,128)(64,129)(65,130)(66,131)(67,132)(68,133)(69,134)(70,135)(71,136)(72,137)(73,138)(74,139)(75,140)(76,141)(77,142)(78,143)(79,144)(80,145), (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), (1,63,110,138,21,43,90,158)(2,50,119,141,22,70,99,121)(3,77,88,144,23,57,108,124)(4,64,97,147,24,44,117,127)(5,51,106,150,25,71,86,130)(6,78,115,153,26,58,95,133)(7,65,84,156,27,45,104,136)(8,52,93,159,28,72,113,139)(9,79,102,122,29,59,82,142)(10,66,111,125,30,46,91,145)(11,53,120,128,31,73,100,148)(12,80,89,131,32,60,109,151)(13,67,98,134,33,47,118,154)(14,54,107,137,34,74,87,157)(15,41,116,140,35,61,96,160)(16,68,85,143,36,48,105,123)(17,55,94,146,37,75,114,126)(18,42,103,149,38,62,83,129)(19,69,112,152,39,49,92,132)(20,56,81,155,40,76,101,135)>;
G:=Group( (1,120)(2,81)(3,82)(4,83)(5,84)(6,85)(7,86)(8,87)(9,88)(10,89)(11,90)(12,91)(13,92)(14,93)(15,94)(16,95)(17,96)(18,97)(19,98)(20,99)(21,100)(22,101)(23,102)(24,103)(25,104)(26,105)(27,106)(28,107)(29,108)(30,109)(31,110)(32,111)(33,112)(34,113)(35,114)(36,115)(37,116)(38,117)(39,118)(40,119)(41,146)(42,147)(43,148)(44,149)(45,150)(46,151)(47,152)(48,153)(49,154)(50,155)(51,156)(52,157)(53,158)(54,159)(55,160)(56,121)(57,122)(58,123)(59,124)(60,125)(61,126)(62,127)(63,128)(64,129)(65,130)(66,131)(67,132)(68,133)(69,134)(70,135)(71,136)(72,137)(73,138)(74,139)(75,140)(76,141)(77,142)(78,143)(79,144)(80,145), (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), (1,63,110,138,21,43,90,158)(2,50,119,141,22,70,99,121)(3,77,88,144,23,57,108,124)(4,64,97,147,24,44,117,127)(5,51,106,150,25,71,86,130)(6,78,115,153,26,58,95,133)(7,65,84,156,27,45,104,136)(8,52,93,159,28,72,113,139)(9,79,102,122,29,59,82,142)(10,66,111,125,30,46,91,145)(11,53,120,128,31,73,100,148)(12,80,89,131,32,60,109,151)(13,67,98,134,33,47,118,154)(14,54,107,137,34,74,87,157)(15,41,116,140,35,61,96,160)(16,68,85,143,36,48,105,123)(17,55,94,146,37,75,114,126)(18,42,103,149,38,62,83,129)(19,69,112,152,39,49,92,132)(20,56,81,155,40,76,101,135) );
G=PermutationGroup([(1,120),(2,81),(3,82),(4,83),(5,84),(6,85),(7,86),(8,87),(9,88),(10,89),(11,90),(12,91),(13,92),(14,93),(15,94),(16,95),(17,96),(18,97),(19,98),(20,99),(21,100),(22,101),(23,102),(24,103),(25,104),(26,105),(27,106),(28,107),(29,108),(30,109),(31,110),(32,111),(33,112),(34,113),(35,114),(36,115),(37,116),(38,117),(39,118),(40,119),(41,146),(42,147),(43,148),(44,149),(45,150),(46,151),(47,152),(48,153),(49,154),(50,155),(51,156),(52,157),(53,158),(54,159),(55,160),(56,121),(57,122),(58,123),(59,124),(60,125),(61,126),(62,127),(63,128),(64,129),(65,130),(66,131),(67,132),(68,133),(69,134),(70,135),(71,136),(72,137),(73,138),(74,139),(75,140),(76,141),(77,142),(78,143),(79,144),(80,145)], [(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)], [(1,63,110,138,21,43,90,158),(2,50,119,141,22,70,99,121),(3,77,88,144,23,57,108,124),(4,64,97,147,24,44,117,127),(5,51,106,150,25,71,86,130),(6,78,115,153,26,58,95,133),(7,65,84,156,27,45,104,136),(8,52,93,159,28,72,113,139),(9,79,102,122,29,59,82,142),(10,66,111,125,30,46,91,145),(11,53,120,128,31,73,100,148),(12,80,89,131,32,60,109,151),(13,67,98,134,33,47,118,154),(14,54,107,137,34,74,87,157),(15,41,116,140,35,61,96,160),(16,68,85,143,36,48,105,123),(17,55,94,146,37,75,114,126),(18,42,103,149,38,62,83,129),(19,69,112,152,39,49,92,132),(20,56,81,155,40,76,101,135)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 14 | 2 | 0 | 0 | 0 | 0 |
| 0 | 38 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 27 | 7 | 34 |
| 0 | 0 | 14 | 27 | 34 | 0 |
| 0 | 0 | 7 | 0 | 34 | 27 |
| 0 | 0 | 14 | 34 | 7 | 27 |
| 38 | 0 | 0 | 0 | 0 | 0 |
| 5 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 36 | 8 |
| 0 | 0 | 0 | 8 | 28 | 13 |
| 0 | 0 | 33 | 13 | 28 | 8 |
| 0 | 0 | 33 | 8 | 36 | 0 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[14,0,0,0,0,0,2,38,0,0,0,0,0,0,0,14,7,14,0,0,27,27,0,34,0,0,7,34,34,7,0,0,34,0,27,27],[38,5,0,0,0,0,0,3,0,0,0,0,0,0,5,0,33,33,0,0,0,8,13,8,0,0,36,28,28,36,0,0,8,13,8,0] >;
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | ··· | 8P | 10A | 10B | 10C | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 2 | 2 | 5 | 5 | 5 | 5 | 4 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | - | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | Q8 | D4 | Q8 | C8.C4 | F5 | C2×F5 | C2×F5 | C4⋊F5 | C4⋊F5 | C40.C4 |
| kernel | C2×C40.C4 | C40.C4 | D5×C2×C8 | C2×C4.F5 | C8×D5 | C2×C5⋊2C8 | C2×C40 | C4×D5 | C4×D5 | C2×Dic5 | C22×D5 | C10 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 4 | 2 | 2 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 2 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_2\times C_{40}.C_4 % in TeX
G:=Group("C2xC40.C4"); // GroupNames label
G:=SmallGroup(320,1060);
// by ID
G=gap.SmallGroup(320,1060);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,100,136,1684,102,6278,1595]);
// Polycyclic
G:=Group<a,b,c|a^2=b^40=1,c^4=b^20,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
×
⋊
ℤ
𝔽
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