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G = C20⋊C8⋊C2order 320 = 26·5

11st semidirect product of C20⋊C8 and C2 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20⋊C811C2, C22⋊C4.5F5, C10.6(C8○D4), C2.9(D4.F5), C23.D5.3C4, C23.26(C2×F5), Dic5⋊C88C2, Dic5.9(C4⋊C4), (C2×Dic5).13Q8, Dic5.11(C2×Q8), Dic5.29(C2×D4), C10.D4.3C4, C22.12(C4⋊F5), (C2×Dic5).114D4, C22.72(C22×F5), C51(C42.6C22), (C2×Dic5).326C23, (C4×Dic5).242C22, C23.11D10.6C2, (C22×Dic5).181C22, C10.7(C2×C4⋊C4), C2.10(C2×C4⋊F5), (C2×C5⋊C8).5C22, (C22×C5⋊C8).3C2, (C2×C4).23(C2×F5), (C2×C10).5(C4⋊C4), (C2×C20).81(C2×C4), (C5×C22⋊C4).3C4, (C2×C22.F5).4C2, (C22×C10).17(C2×C4), (C2×C10).34(C22×C4), (C2×Dic5).51(C2×C4), SmallGroup(320,1034)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C20⋊C8⋊C2
C1C5C10Dic5C2×Dic5C2×C5⋊C8C22×C5⋊C8 — C20⋊C8⋊C2
C5C2×C10 — C20⋊C8⋊C2
C1C22C22⋊C4

Generators and relations for C20⋊C8⋊C2
 G = < a,b,c | a20=b8=c2=1, bab-1=a3, cac=ab4, cbc=b5 >

Subgroups: 346 in 114 conjugacy classes, 52 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×8], C22, C22 [×2], C22 [×2], C5, C8 [×4], C2×C4 [×2], C2×C4 [×8], C23, C10, C10 [×2], C10 [×2], C42 [×2], C22⋊C4, C22⋊C4, C4⋊C4 [×2], C2×C8 [×6], M4(2) [×2], C22×C4, Dic5 [×2], Dic5 [×2], Dic5 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C4⋊C8 [×4], C42⋊C2, C22×C8, C2×M4(2), C5⋊C8 [×4], C2×Dic5 [×2], C2×Dic5 [×6], C2×C20 [×2], C22×C10, C42.6C22, C4×Dic5 [×2], C10.D4 [×2], C23.D5, C5×C22⋊C4, C2×C5⋊C8 [×2], C2×C5⋊C8 [×2], C2×C5⋊C8 [×2], C22.F5 [×2], C22×Dic5, C20⋊C8 [×2], Dic5⋊C8 [×2], C23.11D10, C22×C5⋊C8, C2×C22.F5, C20⋊C8⋊C2
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, F5, C2×C4⋊C4, C8○D4 [×2], C2×F5 [×3], C42.6C22, C4⋊F5 [×2], C22×F5, C2×C4⋊F5, D4.F5 [×2], C20⋊C8⋊C2

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

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J 5 8A···8H8I8J8K8L10A10B10C10D10E20A20B20C20D
order122222444444444458···88888101010101020202020
size11112244555510102020410···1020202020444888888

38 irreducible representations

dim11111111122244448
type+++++++-+++-
imageC1C2C2C2C2C2C4C4C4D4Q8C8○D4F5C2×F5C2×F5C4⋊F5D4.F5
kernelC20⋊C8⋊C2C20⋊C8Dic5⋊C8C23.11D10C22×C5⋊C8C2×C22.F5C10.D4C23.D5C5×C22⋊C4C2×Dic5C2×Dic5C10C22⋊C4C2×C4C23C22C2
# reps12211142222812142

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

0400000
4000000
00734140
00340714
003427147
00727014
,
0380000
3800000
0031213334
0023143724
002741716
007371020
,
100000
0400000
001000
000100
000010
000001

G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,40,0,0,0,0,0,0,0,7,34,34,7,0,0,34,0,27,27,0,0,14,7,14,0,0,0,0,14,7,14],[0,38,0,0,0,0,38,0,0,0,0,0,0,0,31,23,27,7,0,0,21,14,4,37,0,0,33,37,17,10,0,0,34,24,16,20],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C20⋊C8⋊C2 in GAP, Magma, Sage, TeX

C_{20}\rtimes C_8\rtimes C_2
% in TeX

G:=Group("C20:C8:C2");
// GroupNames label

G:=SmallGroup(320,1034);
// by ID

G=gap.SmallGroup(320,1034);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,387,100,136,6278,1595]);
// Polycyclic

G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^3,c*a*c=a*b^4,c*b*c=b^5>;
// generators/relations

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