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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, C2, C4, C22, C22, C5, C8, C8, C2×C4, C23, C10, C10, C10, C2×C8, C2×C8, C2×C8, M4(2), C22×C4, C20, C2×C10, C2×C10, C8.C4, C22×C8, C2×M4(2), C52C8, C40, C2×C20, C22×C10, C2×C8.C4, C2×C52C8, C4.Dic5, C4.Dic5, C2×C40, C2×C40, C22×C20, C40.6C4, C2×C4.Dic5, C22×C40, C2×C40.6C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic5, D10, C8.C4, C2×C4⋊C4, Dic10, D20, C2×Dic5, C22×D5, C2×C8.C4, C4⋊Dic5, C2×Dic10, C2×D20, C22×Dic5, C40.6C4, C2×C4⋊Dic5, C2×C40.6C4

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

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

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

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

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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