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G = (C2×C20)⋊C8order 320 = 26·5

6th semidirect product of C2×C20 and C8 acting via C8/C2=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C20)⋊6C8, (C2×C20).228D4, (C22×C20).2C4, (C22×C4).4D10, (C22×C4).3Dic5, C10.29(C22⋊C8), C10.39(C23⋊C4), (C2×C10).41M4(2), C2.3(C23⋊Dic5), C23.22(C2×Dic5), C20.55D4.12C2, C2.4(C20.55D4), C2.1(C20.10D4), C22.5(C4.Dic5), C10.11(C4.10D4), (C22×C20).325C22, C54(C22.M4(2)), C22.25(C23.D5), (C2×C4)⋊(C52C8), (C2×C4⋊C4).1D5, (C10×C4⋊C4).25C2, (C2×C10).58(C2×C8), C22.3(C2×C52C8), (C2×C4).160(C5⋊D4), (C22×C10).192(C2×C4), (C2×C10).150(C22⋊C4), SmallGroup(320,86)

Series: Derived Chief Lower central Upper central

C1C2×C10 — (C2×C20)⋊C8
C1C5C10C2×C10C2×C20C22×C20C20.55D4 — (C2×C20)⋊C8
C5C10C2×C10 — (C2×C20)⋊C8
C1C22C22×C4C2×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 [×3], C2 [×2], C4 [×5], C22 [×3], C22 [×2], C5, C8 [×2], C2×C4 [×4], C2×C4 [×5], C23, C10 [×3], C10 [×2], C4⋊C4 [×2], C2×C8 [×2], C22×C4 [×3], C20 [×5], C2×C10 [×3], C2×C10 [×2], C22⋊C8 [×2], C2×C4⋊C4, C52C8 [×2], C2×C20 [×4], C2×C20 [×5], C22×C10, C22.M4(2), C2×C52C8 [×2], C5×C4⋊C4 [×2], C22×C20 [×3], C20.55D4 [×2], C10×C4⋊C4, (C2×C20)⋊C8
Quotients: C1, C2 [×3], C4 [×2], C22, C8 [×2], C2×C4, D4 [×2], D5, C22⋊C4, C2×C8, M4(2), Dic5 [×2], D10, C22⋊C8, C23⋊C4, C4.10D4, C52C8 [×2], C2×Dic5, C5⋊D4 [×2], 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)(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)
(1 160 43 136 38 107 84 62)(2 159 54 125 39 106 95 71)(3 158 45 134 40 105 86 80)(4 157 56 123 21 104 97 69)(5 156 47 132 22 103 88 78)(6 155 58 121 23 102 99 67)(7 154 49 130 24 101 90 76)(8 153 60 139 25 120 81 65)(9 152 51 128 26 119 92 74)(10 151 42 137 27 118 83 63)(11 150 53 126 28 117 94 72)(12 149 44 135 29 116 85 61)(13 148 55 124 30 115 96 70)(14 147 46 133 31 114 87 79)(15 146 57 122 32 113 98 68)(16 145 48 131 33 112 89 77)(17 144 59 140 34 111 100 66)(18 143 50 129 35 110 91 75)(19 142 41 138 36 109 82 64)(20 141 52 127 37 108 93 73)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H5A5B8A···8H10A···10N20A···20X
order12222244444444558···810···1020···20
size111122222244442220···202···24···4

62 irreducible representations

dim11111222222224444
type+++++-++-
imageC1C2C2C4C8D4D5M4(2)Dic5D10C52C8C5⋊D4C4.Dic5C23⋊C4C4.10D4C23⋊Dic5C20.10D4
kernel(C2×C20)⋊C8C20.55D4C10×C4⋊C4C22×C20C2×C20C2×C20C2×C4⋊C4C2×C10C22×C4C22×C4C2×C4C2×C4C22C10C10C2C2
# reps12148222428881144

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

4000000
0400000
0010031
0001031
0000400
0000040
,
010000
100000
0036293033
001253027
0000356
000066
,
0380000
300000
00310129
00310029
0040100
00390010

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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