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G = C88D20order 320 = 26·5

2nd semidirect product of C8 and D20 acting via D20/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C88D20, C4013D4, D103SD16, C4.Q88D5, C53(C88D4), C4⋊C4.38D10, C4.50(C2×D20), D102Q86C2, (C2×C8).260D10, C4⋊D20.6C2, C20.130(C2×D4), D206C415C2, C2.24(D5×SD16), C10.55(C4○D8), C20.29(C4○D4), C10.Q1616C2, C4.3(Q82D5), C10.40(C2×SD16), (C22×D5).84D4, C22.216(D4×D5), C10.43(C4⋊D4), C2.16(C4⋊D20), (C2×C40).161C22, (C2×C20).280C23, (C2×Dic5).144D4, (C2×D20).78C22, C2.22(SD163D5), (C2×Dic10).87C22, (D5×C2×C8)⋊7C2, (C5×C4.Q8)⋊9C2, (C2×C40⋊C2)⋊28C2, (C2×C10).285(C2×D4), (C5×C4⋊C4).73C22, (C2×C4×D5).303C22, (C2×C4).383(C22×D5), (C2×C52C8).236C22, SmallGroup(320,491)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C88D20
C1C5C10C2×C10C2×C20C2×C4×D5D5×C2×C8 — C88D20
C5C10C2×C20 — C88D20
C1C22C2×C4C4.Q8

Generators and relations for C88D20
 G = < a,b,c | a8=b20=c2=1, bab-1=cac=a3, cbc=b-1 >

Subgroups: 598 in 124 conjugacy classes, 43 normal (37 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×7], C5, C8 [×2], C8, C2×C4, C2×C4 [×6], D4 [×4], Q8 [×2], C23 [×2], D5 [×3], C10 [×3], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4, C2×C8, C2×C8 [×3], SD16 [×2], C22×C4, C2×D4 [×2], C2×Q8, Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×5], C2×C10, D4⋊C4, Q8⋊C4, C4.Q8, C4⋊D4, C22⋊Q8, C22×C8, C2×SD16, C52C8, C40 [×2], Dic10 [×2], C4×D5 [×2], D20 [×4], C2×Dic5, C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, C22×D5, C88D4, C8×D5 [×2], C40⋊C2 [×2], C2×C52C8, C4⋊Dic5, D10⋊C4 [×2], C5×C4⋊C4 [×2], C2×C40, C2×Dic10, C2×C4×D5, C2×D20, C2×D20, D206C4, C10.Q16, C5×C4.Q8, C4⋊D20, D102Q8, D5×C2×C8, C2×C40⋊C2, C88D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, SD16 [×2], C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C2×SD16, C4○D8, D20 [×2], C22×D5, C88D4, C2×D20, D4×D5, Q82D5, C4⋊D20, D5×SD16, SD163D5, C88D20

Smallest permutation representation of C88D20
On 160 points
Generators in S160
(1 125 117 91 30 151 79 59)(2 92 80 126 31 60 118 152)(3 127 119 93 32 153 61 41)(4 94 62 128 33 42 120 154)(5 129 101 95 34 155 63 43)(6 96 64 130 35 44 102 156)(7 131 103 97 36 157 65 45)(8 98 66 132 37 46 104 158)(9 133 105 99 38 159 67 47)(10 100 68 134 39 48 106 160)(11 135 107 81 40 141 69 49)(12 82 70 136 21 50 108 142)(13 137 109 83 22 143 71 51)(14 84 72 138 23 52 110 144)(15 139 111 85 24 145 73 53)(16 86 74 140 25 54 112 146)(17 121 113 87 26 147 75 55)(18 88 76 122 27 56 114 148)(19 123 115 89 28 149 77 57)(20 90 78 124 29 58 116 150)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 33)(22 32)(23 31)(24 30)(25 29)(26 28)(34 40)(35 39)(36 38)(41 143)(42 142)(43 141)(44 160)(45 159)(46 158)(47 157)(48 156)(49 155)(50 154)(51 153)(52 152)(53 151)(54 150)(55 149)(56 148)(57 147)(58 146)(59 145)(60 144)(61 109)(62 108)(63 107)(64 106)(65 105)(66 104)(67 103)(68 102)(69 101)(70 120)(71 119)(72 118)(73 117)(74 116)(75 115)(76 114)(77 113)(78 112)(79 111)(80 110)(81 129)(82 128)(83 127)(84 126)(85 125)(86 124)(87 123)(88 122)(89 121)(90 140)(91 139)(92 138)(93 137)(94 136)(95 135)(96 134)(97 133)(98 132)(99 131)(100 130)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G5A5B8A8B8C8D8E8F8G8H10A···10F20A20B20C20D20E···20L40A···40H
order12222224444444558888888810···102020202020···2040···40
size11111010402288101040222222101010102···244448···84···4

50 irreducible representations

dim1111111122222222224444
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D5C4○D4SD16D10D10C4○D8D20Q82D5D4×D5D5×SD16SD163D5
kernelC88D20D206C4C10.Q16C5×C4.Q8C4⋊D20D102Q8D5×C2×C8C2×C40⋊C2C40C2×Dic5C22×D5C4.Q8C20D10C4⋊C4C2×C8C10C8C4C22C2C2
# reps1111111121122442482244

Matrix representation of C88D20 in GL6(𝔽41)

100000
010000
001000
000100
0000030
00002630
,
010000
40340000
001500
00164000
000010
0000140
,
010000
100000
0040000
0025100
000010
0000140

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,26,0,0,0,0,30,30],[0,40,0,0,0,0,1,34,0,0,0,0,0,0,1,16,0,0,0,0,5,40,0,0,0,0,0,0,1,1,0,0,0,0,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,25,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,40] >;

C88D20 in GAP, Magma, Sage, TeX

C_8\rtimes_8D_{20}
% in TeX

G:=Group("C8:8D20");
// GroupNames label

G:=SmallGroup(320,491);
// by ID

G=gap.SmallGroup(320,491);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,555,58,438,102,12550]);
// Polycyclic

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

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