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## G = C2×C20.C8order 320 = 26·5

### Direct product of C2 and C20.C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×C20.C8
 Chief series C1 — C5 — C10 — C20 — C5⋊2C8 — C5⋊C16 — C2×C5⋊C16 — C2×C20.C8
 Lower central C5 — C10 — C2×C20.C8
 Upper central C1 — C2×C4 — C22×C4

Generators and relations for C2×C20.C8
G = < a,b,c | a2=b20=1, c8=b10, ab=ba, ac=ca, cbc-1=b3 >

Subgroups: 186 in 90 conjugacy classes, 60 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×2], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C23, C10, C10 [×2], C10 [×2], C16 [×4], C2×C8 [×6], C22×C4, C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C16 [×2], M5(2) [×4], C22×C8, C52C8 [×2], C52C8 [×2], C2×C20 [×2], C2×C20 [×4], C22×C10, C2×M5(2), C5⋊C16 [×4], C2×C52C8 [×2], C2×C52C8 [×4], C22×C20, C2×C5⋊C16 [×2], C20.C8 [×4], C22×C52C8, C2×C20.C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, C2×C8 [×6], C22×C4, F5, M5(2) [×2], C22×C8, C5⋊C8 [×4], C2×F5 [×3], C2×M5(2), C2×C5⋊C8 [×6], C22×F5, C20.C8 [×2], C22×C5⋊C8, C2×C20.C8

Smallest permutation representation of C2×C20.C8
On 160 points
Generators in S160
(1 150)(2 151)(3 152)(4 153)(5 154)(6 155)(7 156)(8 157)(9 158)(10 159)(11 160)(12 141)(13 142)(14 143)(15 144)(16 145)(17 146)(18 147)(19 148)(20 149)(21 56)(22 57)(23 58)(24 59)(25 60)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(61 102)(62 103)(63 104)(64 105)(65 106)(66 107)(67 108)(68 109)(69 110)(70 111)(71 112)(72 113)(73 114)(74 115)(75 116)(76 117)(77 118)(78 119)(79 120)(80 101)(81 126)(82 127)(83 128)(84 129)(85 130)(86 131)(87 132)(88 133)(89 134)(90 135)(91 136)(92 137)(93 138)(94 139)(95 140)(96 121)(97 122)(98 123)(99 124)(100 125)
(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 92 30 65 155 132 50 101 11 82 40 75 145 122 60 111)(2 99 39 68 156 139 59 104 12 89 29 78 146 129 49 114)(3 86 28 71 157 126 48 107 13 96 38 61 147 136 58 117)(4 93 37 74 158 133 57 110 14 83 27 64 148 123 47 120)(5 100 26 77 159 140 46 113 15 90 36 67 149 130 56 103)(6 87 35 80 160 127 55 116 16 97 25 70 150 137 45 106)(7 94 24 63 141 134 44 119 17 84 34 73 151 124 54 109)(8 81 33 66 142 121 53 102 18 91 23 76 152 131 43 112)(9 88 22 69 143 128 42 105 19 98 32 79 153 138 52 115)(10 95 31 72 144 135 51 108 20 85 21 62 154 125 41 118)

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 5 8A ··· 8H 8I 8J 8K 8L 10A ··· 10G 16A ··· 16P 20A ··· 20H order 1 2 2 2 2 2 4 4 4 4 4 4 5 8 ··· 8 8 8 8 8 10 ··· 10 16 ··· 16 20 ··· 20 size 1 1 1 1 2 2 1 1 1 1 2 2 4 5 ··· 5 10 10 10 10 4 ··· 4 10 ··· 10 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 4 4 4 4 4 type + + + + + - + - image C1 C2 C2 C2 C4 C4 C8 C8 M5(2) F5 C5⋊C8 C2×F5 C5⋊C8 C20.C8 kernel C2×C20.C8 C2×C5⋊C16 C20.C8 C22×C5⋊2C8 C2×C5⋊2C8 C22×C20 C2×C20 C22×C10 C10 C22×C4 C2×C4 C2×C4 C23 C2 # reps 1 2 4 1 6 2 12 4 8 1 3 3 1 8

Matrix representation of C2×C20.C8 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 177 0 0 0 0 0 236 64 0 0 0 0 0 0 131 64 0 0 0 0 177 0 0 0 0 0 116 117 0 110 0 0 58 7 195 195
,
 177 96 0 0 0 0 43 64 0 0 0 0 0 0 0 0 190 240 0 0 64 64 2 1 0 0 83 185 177 0 0 0 207 215 131 0

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[177,236,0,0,0,0,0,64,0,0,0,0,0,0,131,177,116,58,0,0,64,0,117,7,0,0,0,0,0,195,0,0,0,0,110,195],[177,43,0,0,0,0,96,64,0,0,0,0,0,0,0,64,83,207,0,0,0,64,185,215,0,0,190,2,177,131,0,0,240,1,0,0] >;

C2×C20.C8 in GAP, Magma, Sage, TeX

C_2\times C_{20}.C_8
% in TeX

G:=Group("C2xC20.C8");
// GroupNames label

G:=SmallGroup(320,1081);
// by ID

G=gap.SmallGroup(320,1081);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,80,102,6278,1595]);
// Polycyclic

G:=Group<a,b,c|a^2=b^20=1,c^8=b^10,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations

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