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## G = (C2×C20)⋊C8order 320 = 26·5

### 6th 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)⋊C8
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C22×C20 — C20.55D4 — (C2×C20)⋊C8
 Lower central C5 — C10 — C2×C10 — (C2×C20)⋊C8
 Upper central C1 — C22 — C22×C4 — C2×C4⋊C4

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

Subgroups: 198 in 78 conjugacy classes, 35 normal (25 characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C4⋊C4, C2×C8, C22×C4, C20, C2×C10, C2×C10, C22⋊C8, C2×C4⋊C4, C52C8, C2×C20, C2×C20, C22×C10, C22.M4(2), C2×C52C8, C5×C4⋊C4, C22×C20, C20.55D4, C10×C4⋊C4, (C2×C20)⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, D5, C22⋊C4, C2×C8, M4(2), Dic5, D10, C22⋊C8, C23⋊C4, C4.10D4, C52C8, C2×Dic5, C5⋊D4, C22.M4(2), C2×C52C8, C4.Dic5, C23.D5, C20.55D4, C23⋊Dic5, C20.10D4, (C2×C20)⋊C8

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

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

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

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

62 conjugacy classes

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

62 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + - + + - image C1 C2 C2 C4 C8 D4 D5 M4(2) Dic5 D10 C5⋊2C8 C5⋊D4 C4.Dic5 C23⋊C4 C4.10D4 C23⋊Dic5 C20.10D4 kernel (C2×C20)⋊C8 C20.55D4 C10×C4⋊C4 C22×C20 C2×C20 C2×C20 C2×C4⋊C4 C2×C10 C22×C4 C22×C4 C2×C4 C2×C4 C22 C10 C10 C2 C2 # reps 1 2 1 4 8 2 2 2 4 2 8 8 8 1 1 4 4

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

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 31 0 0 0 1 0 31 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 36 29 30 33 0 0 12 5 30 27 0 0 0 0 35 6 0 0 0 0 6 6
,
 0 38 0 0 0 0 3 0 0 0 0 0 0 0 31 0 1 29 0 0 31 0 0 29 0 0 40 1 0 0 0 0 39 0 0 10

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,31,31,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,36,12,0,0,0,0,29,5,0,0,0,0,30,30,35,6,0,0,33,27,6,6],[0,3,0,0,0,0,38,0,0,0,0,0,0,0,31,31,40,39,0,0,0,0,1,0,0,0,1,0,0,0,0,0,29,29,0,10] >;

(C2×C20)⋊C8 in GAP, Magma, Sage, TeX

(C_2\times C_{20})\rtimes C_8
% in TeX

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

G:=SmallGroup(320,86);
// by ID

G=gap.SmallGroup(320,86);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,387,100,1123,12550]);
// 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^-1>;
// generators/relations

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