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## G = C2×C40.6C4order 320 = 26·5

### Direct product of C2 and C40.6C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×C40.6C4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4.Dic5 — C2×C4.Dic5 — C2×C40.6C4
 Lower central C5 — C10 — C20 — C2×C40.6C4
 Upper central C1 — C2×C4 — C22×C4 — C22×C8

Generators and relations for C2×C40.6C4
G = < a,b,c | a2=b40=1, c4=b20, ab=ba, ac=ca, cbc-1=b19 >

Subgroups: 238 in 106 conjugacy classes, 71 normal (41 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C22 [×3], C22 [×2], C5, C8 [×4], C8 [×4], C2×C4 [×6], C23, C10, C10 [×2], C10 [×2], C2×C8 [×2], C2×C8 [×4], C2×C8 [×2], M4(2) [×6], C22×C4, C20 [×4], C2×C10 [×3], C2×C10 [×2], C8.C4 [×4], C22×C8, C2×M4(2) [×2], C52C8 [×4], C40 [×4], C2×C20 [×6], C22×C10, C2×C8.C4, C2×C52C8 [×2], C4.Dic5 [×4], C4.Dic5 [×2], C2×C40 [×2], C2×C40 [×4], C22×C20, C40.6C4 [×4], C2×C4.Dic5 [×2], C22×C40, C2×C40.6C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, Dic5 [×4], D10 [×3], C8.C4 [×2], C2×C4⋊C4, Dic10 [×2], D20 [×2], C2×Dic5 [×6], C22×D5, C2×C8.C4, C4⋊Dic5 [×4], C2×Dic10, C2×D20, C22×Dic5, C40.6C4 [×2], C2×C4⋊Dic5, C2×C40.6C4

Smallest permutation representation of C2×C40.6C4
On 160 points
Generators in S160
(1 97)(2 98)(3 99)(4 100)(5 101)(6 102)(7 103)(8 104)(9 105)(10 106)(11 107)(12 108)(13 109)(14 110)(15 111)(16 112)(17 113)(18 114)(19 115)(20 116)(21 117)(22 118)(23 119)(24 120)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 133)(42 134)(43 135)(44 136)(45 137)(46 138)(47 139)(48 140)(49 141)(50 142)(51 143)(52 144)(53 145)(54 146)(55 147)(56 148)(57 149)(58 150)(59 151)(60 152)(61 153)(62 154)(63 155)(64 156)(65 157)(66 158)(67 159)(68 160)(69 121)(70 122)(71 123)(72 124)(73 125)(74 126)(75 127)(76 128)(77 129)(78 130)(79 131)(80 132)
(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 43 107 125 21 63 87 145)(2 62 108 144 22 42 88 124)(3 41 109 123 23 61 89 143)(4 60 110 142 24 80 90 122)(5 79 111 121 25 59 91 141)(6 58 112 140 26 78 92 160)(7 77 113 159 27 57 93 139)(8 56 114 138 28 76 94 158)(9 75 115 157 29 55 95 137)(10 54 116 136 30 74 96 156)(11 73 117 155 31 53 97 135)(12 52 118 134 32 72 98 154)(13 71 119 153 33 51 99 133)(14 50 120 132 34 70 100 152)(15 69 81 151 35 49 101 131)(16 48 82 130 36 68 102 150)(17 67 83 149 37 47 103 129)(18 46 84 128 38 66 104 148)(19 65 85 147 39 45 105 127)(20 44 86 126 40 64 106 146)

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

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

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

92 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 5A 5B 8A ··· 8H 8I ··· 8P 10A ··· 10N 20A ··· 20P 40A ··· 40AF order 1 2 2 2 2 2 4 4 4 4 4 4 5 5 8 ··· 8 8 ··· 8 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 1 1 2 2 1 1 1 1 2 2 2 2 2 ··· 2 20 ··· 20 2 ··· 2 2 ··· 2 2 ··· 2

92 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + - - + - + + - + - image C1 C2 C2 C2 C4 D4 Q8 Q8 D5 Dic5 D10 D10 C8.C4 Dic10 D20 Dic10 C40.6C4 kernel C2×C40.6C4 C40.6C4 C2×C4.Dic5 C22×C40 C2×C40 C2×C20 C2×C20 C22×C10 C22×C8 C2×C8 C2×C8 C22×C4 C10 C2×C4 C2×C4 C23 C2 # reps 1 4 2 1 8 2 1 1 2 8 4 2 8 4 8 4 32

Matrix representation of C2×C40.6C4 in GL3(𝔽41) generated by

 40 0 0 0 40 0 0 0 40
,
 1 0 0 0 19 0 0 0 28
,
 40 0 0 0 0 27 0 14 0
G:=sub<GL(3,GF(41))| [40,0,0,0,40,0,0,0,40],[1,0,0,0,19,0,0,0,28],[40,0,0,0,0,14,0,27,0] >;

C2×C40.6C4 in GAP, Magma, Sage, TeX

C_2\times C_{40}._6C_4
% in TeX

G:=Group("C2xC40.6C4");
// GroupNames label

G:=SmallGroup(320,734);
// by ID

G=gap.SmallGroup(320,734);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,100,136,1684,102,12550]);
// 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^19>;
// generators/relations

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