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## G = (C2×C20)⋊1C8order 320 = 26·5

### 1st semidirect product of C2×C20 and C8 acting via C8/C2=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — (C2×C20)⋊1C8
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C23.2F5 — (C2×C20)⋊1C8
 Lower central C5 — C10 — C2×C10 — (C2×C20)⋊1C8
 Upper central C1 — C22 — C23 — C22×C4

Generators and relations for (C2×C20)⋊1C8
G = < a,b,c | a2=b20=c8=1, ab=ba, cac-1=ab10, cbc-1=ab3 >

Subgroups: 306 in 78 conjugacy classes, 30 normal (22 characteristic)
C1, C2 [×3], C2 [×2], C4 [×5], C22 [×3], C22 [×2], C5, C8 [×2], C2×C4 [×2], C2×C4 [×7], C23, C10 [×3], C10 [×2], C4⋊C4 [×2], C2×C8 [×2], C22×C4, C22×C4 [×2], Dic5 [×3], C20 [×2], C2×C10 [×3], C2×C10 [×2], C22⋊C8 [×2], C2×C4⋊C4, C5⋊C8 [×2], C2×Dic5 [×2], C2×Dic5 [×3], C2×C20 [×2], C2×C20 [×2], C22×C10, C22.M4(2), C4⋊Dic5 [×2], C2×C5⋊C8 [×2], C22×Dic5 [×2], C22×C20, C23.2F5 [×2], C2×C4⋊Dic5, (C2×C20)⋊1C8
Quotients: C1, C2 [×3], C4 [×2], C22, C8 [×2], C2×C4, D4 [×2], C22⋊C4, C2×C8, M4(2), F5, C22⋊C8, C23⋊C4, C4.10D4, C5⋊C8 [×2], C2×F5, C22.M4(2), C2×C5⋊C8, C22.F5, C22⋊F5, D10.D4, Dic5.D4, C23.2F5, (C2×C20)⋊1C8

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

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 5 8A ··· 8H 10A ··· 10G 20A ··· 20H order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 5 8 ··· 8 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 4 4 10 10 10 10 20 20 4 20 ··· 20 4 ··· 4 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 2 2 4 4 4 4 4 4 4 4 4 type + + + + + + - - + - + + - image C1 C2 C2 C4 C4 C8 D4 M4(2) F5 C23⋊C4 C4.10D4 C5⋊C8 C2×F5 C22.F5 C22⋊F5 D10.D4 Dic5.D4 kernel (C2×C20)⋊1C8 C23.2F5 C2×C4⋊Dic5 C22×Dic5 C22×C20 C2×C20 C2×Dic5 C2×C10 C22×C4 C10 C10 C2×C4 C23 C22 C22 C2 C2 # reps 1 2 1 2 2 8 2 2 1 1 1 2 1 2 2 4 4

Matrix representation of (C2×C20)⋊1C8 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 25 39 0 0 0 0 2 13 0 0 0 0 0 0 25 16 0 0 0 0 25 39
,
 0 3 0 0 0 0 38 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 25 39 0 0 0 0 25 16 0 0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,25,2,0,0,0,0,39,13,0,0,0,0,0,0,25,25,0,0,0,0,16,39],[0,38,0,0,0,0,3,0,0,0,0,0,0,0,0,0,25,25,0,0,0,0,39,16,0,0,1,0,0,0,0,0,0,1,0,0] >;

(C2×C20)⋊1C8 in GAP, Magma, Sage, TeX

(C_2\times C_{20})\rtimes_1C_8
% in TeX

G:=Group("(C2xC20):1C8");
// GroupNames label

G:=SmallGroup(320,251);
// by ID

G=gap.SmallGroup(320,251);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,387,100,1123,6278,3156]);
// Polycyclic

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

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