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G = C86D20order 320 = 26·5

3rd semidirect product of C8 and D20 acting via D20/C20=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C86D20, C4025D4, C2010M4(2), C42.257D10, (C4×C8)⋊14D5, (C4×C40)⋊21C2, C54(C86D4), C203C82C2, C2.8(C4×D20), C42(C8⋊D5), (C4×D20).2C2, C4.75(C2×D20), C10.35(C4×D4), D101C81C2, (C2×D20).17C4, C20.295(C2×D4), (C2×C8).283D10, C4⋊Dic5.19C4, C10.28(C8○D4), D10⋊C4.11C4, C4.126(C4○D20), C20.242(C4○D4), (C4×C20).324C22, (C2×C40).343C22, (C2×C20).806C23, C10.38(C2×M4(2)), C2.7(D20.3C4), C2.7(C2×C8⋊D5), (C2×C8⋊D5)⋊10C2, C22.95(C2×C4×D5), (C2×C4).104(C4×D5), (C2×C20).394(C2×C4), (C2×C4×D5).226C22, (C2×Dic5).13(C2×C4), (C22×D5).14(C2×C4), (C2×C4).748(C22×D5), (C2×C10).162(C22×C4), (C2×C52C8).191C22, SmallGroup(320,315)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C86D20
C1C5C10C20C2×C20C2×C4×D5C4×D20 — C86D20
C5C2×C10 — C86D20
C1C2×C4C4×C8

Generators and relations for C86D20
 G = < a,b,c | a8=b20=c2=1, ab=ba, cac=a5, cbc=b-1 >

Subgroups: 446 in 122 conjugacy classes, 55 normal (33 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×3], C22, C22 [×6], C5, C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×6], D4 [×2], C23 [×2], D5 [×2], C10 [×3], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8 [×2], M4(2) [×4], C22×C4 [×2], C2×D4, Dic5 [×2], C20 [×2], C20 [×2], C20, D10 [×6], C2×C10, C4×C8, C22⋊C8 [×2], C4⋊C8, C4×D4, C2×M4(2) [×2], C52C8 [×2], C40 [×2], C40, C4×D5 [×4], D20 [×2], C2×Dic5 [×2], C2×C20 [×3], C22×D5 [×2], C86D4, C8⋊D5 [×4], C2×C52C8 [×2], C4⋊Dic5, D10⋊C4 [×2], C4×C20, C2×C40 [×2], C2×C4×D5 [×2], C2×D20, C203C8, D101C8 [×2], C4×C40, C4×D20, C2×C8⋊D5 [×2], C86D20
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, M4(2) [×2], C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C2×M4(2), C8○D4, C4×D5 [×2], D20 [×2], C22×D5, C86D4, C8⋊D5 [×2], C2×C4×D5, C2×D20, C4○D20, C4×D20, C2×C8⋊D5, D20.3C4, C86D20

Smallest permutation representation of C86D20
On 160 points
Generators in S160
(1 68 103 148 138 41 87 25)(2 69 104 149 139 42 88 26)(3 70 105 150 140 43 89 27)(4 71 106 151 121 44 90 28)(5 72 107 152 122 45 91 29)(6 73 108 153 123 46 92 30)(7 74 109 154 124 47 93 31)(8 75 110 155 125 48 94 32)(9 76 111 156 126 49 95 33)(10 77 112 157 127 50 96 34)(11 78 113 158 128 51 97 35)(12 79 114 159 129 52 98 36)(13 80 115 160 130 53 99 37)(14 61 116 141 131 54 100 38)(15 62 117 142 132 55 81 39)(16 63 118 143 133 56 82 40)(17 64 119 144 134 57 83 21)(18 65 120 145 135 58 84 22)(19 66 101 146 136 59 85 23)(20 67 102 147 137 60 86 24)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 151)(22 150)(23 149)(24 148)(25 147)(26 146)(27 145)(28 144)(29 143)(30 142)(31 141)(32 160)(33 159)(34 158)(35 157)(36 156)(37 155)(38 154)(39 153)(40 152)(41 67)(42 66)(43 65)(44 64)(45 63)(46 62)(47 61)(48 80)(49 79)(50 78)(51 77)(52 76)(53 75)(54 74)(55 73)(56 72)(57 71)(58 70)(59 69)(60 68)(81 92)(82 91)(83 90)(84 89)(85 88)(86 87)(93 100)(94 99)(95 98)(96 97)(101 104)(102 103)(105 120)(106 119)(107 118)(108 117)(109 116)(110 115)(111 114)(112 113)(121 134)(122 133)(123 132)(124 131)(125 130)(126 129)(127 128)(135 140)(136 139)(137 138)

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

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

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

92 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J5A5B8A···8H8I8J8K8L10A···10F20A···20X40A···40AF
order1222224444444444558···8888810···1020···2040···40
size11112020111122222020222···2202020202···22···22···2

92 irreducible representations

dim111111111222222222222
type+++++++++++
imageC1C2C2C2C2C2C4C4C4D4D5M4(2)C4○D4D10D10C8○D4D20C4×D5C8⋊D5C4○D20D20.3C4
kernelC86D20C203C8D101C8C4×C40C4×D20C2×C8⋊D5C4⋊Dic5D10⋊C4C2×D20C40C4×C8C20C20C42C2×C8C10C8C2×C4C4C4C2
# reps11211224222422448816816

Matrix representation of C86D20 in GL4(𝔽41) generated by

9000
0900
00352
00396
,
273000
113200
00040
00135
,
111400
93000
00040
00400
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,35,39,0,0,2,6],[27,11,0,0,30,32,0,0,0,0,0,1,0,0,40,35],[11,9,0,0,14,30,0,0,0,0,0,40,0,0,40,0] >;

C86D20 in GAP, Magma, Sage, TeX

C_8\rtimes_6D_{20}
% in TeX

G:=Group("C8:6D20");
// GroupNames label

G:=SmallGroup(320,315);
// by ID

G=gap.SmallGroup(320,315);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,758,58,136,12550]);
// Polycyclic

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

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