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## G = C2×C40.C4order 320 = 26·5

### Direct product of C2 and C40.C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×C40.C4
 Chief series C1 — C5 — C10 — Dic5 — C4×D5 — C4.F5 — C2×C4.F5 — C2×C40.C4
 Lower central C5 — C10 — C20 — C2×C40.C4
 Upper central C1 — C22 — C2×C4 — C2×C8

Generators and relations for C2×C40.C4
G = < a,b,c | a2=b40=1, c4=b20, ab=ba, ac=ca, cbc-1=b3 >

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], C52C8 [×2], C40 [×2], C5⋊C8 [×4], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, C2×C8.C4, C8×D5 [×4], C2×C52C8, 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

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

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

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

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

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

Matrix representation of C2×C40.C4 in 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] >;

C2×C40.C4 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

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