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G = C5⋊C87D4order 320 = 26·5

4th semidirect product of C5⋊C8 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C5⋊C87D4, C54(C89D4), (C2×D4).5F5, C2.27(D4×F5), (D4×C10).4C4, C10.27(C4×D4), C4⋊Dic5.7C4, (C2×C10)⋊2M4(2), C23.D5.6C4, C23.11(C2×F5), Dic5⋊C83C2, C10.15(C8○D4), Dic5.79(C2×D4), (D4×Dic5).10C2, C2.15(D4.F5), C23.2F59C2, C10.C424C2, C10.30(C2×M4(2)), C221(C22.F5), Dic5.58(C4○D4), C22.91(C22×F5), (C4×Dic5).69C22, (C2×Dic5).352C23, (C22×Dic5).185C22, (C22×C5⋊C8)⋊5C2, (C2×C4).36(C2×F5), (C2×C20).23(C2×C4), (C2×C5⋊C8).40C22, (C2×C22.F5)⋊4C2, C2.9(C2×C22.F5), (C22×C10).24(C2×C4), (C2×C10).76(C22×C4), (C2×Dic5).71(C2×C4), SmallGroup(320,1111)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C5⋊C87D4
C1C5C10Dic5C2×Dic5C2×C5⋊C8C22×C5⋊C8 — C5⋊C87D4
C5C2×C10 — C5⋊C87D4
C1C22C2×D4

Generators and relations for C5⋊C87D4
 G = < a,b,c,d | a5=b8=c4=d2=1, bab-1=a3, ac=ca, ad=da, cbc-1=b5, bd=db, dcd=c-1 >

Subgroups: 394 in 124 conjugacy classes, 48 normal (42 characteristic)
C1, C2 [×3], C2 [×3], C4 [×6], C22, C22 [×2], C22 [×5], C5, C8 [×5], C2×C4, C2×C4 [×8], D4 [×2], C23 [×2], C10 [×3], C10 [×3], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×6], M4(2) [×2], C22×C4 [×2], C2×D4, Dic5 [×2], Dic5 [×3], C20, C2×C10, C2×C10 [×2], C2×C10 [×5], C8⋊C4, C22⋊C8 [×2], C4⋊C8, C4×D4, C22×C8, C2×M4(2), C5⋊C8 [×2], C5⋊C8 [×3], C2×Dic5 [×4], C2×Dic5 [×4], C2×C20, C5×D4 [×2], C22×C10 [×2], C89D4, C4×Dic5, C4⋊Dic5, C23.D5 [×2], C2×C5⋊C8 [×4], C2×C5⋊C8 [×2], C22.F5 [×2], C22×Dic5 [×2], D4×C10, C10.C42, Dic5⋊C8, C23.2F5 [×2], D4×Dic5, C22×C5⋊C8, C2×C22.F5, C5⋊C87D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, M4(2) [×2], C22×C4, C2×D4, C4○D4, F5, C4×D4, C2×M4(2), C8○D4, C2×F5 [×3], C89D4, C22.F5 [×2], C22×F5, D4.F5, D4×F5, C2×C22.F5, C5⋊C87D4

Smallest permutation representation of C5⋊C87D4
On 160 points
Generators in S160
(1 65 83 36 159)(2 37 66 160 84)(3 153 38 85 67)(4 86 154 68 39)(5 69 87 40 155)(6 33 70 156 88)(7 157 34 81 71)(8 82 158 72 35)(9 133 122 54 103)(10 55 134 104 123)(11 97 56 124 135)(12 125 98 136 49)(13 129 126 50 99)(14 51 130 100 127)(15 101 52 128 131)(16 121 102 132 53)(17 138 114 90 63)(18 91 139 64 115)(19 57 92 116 140)(20 117 58 141 93)(21 142 118 94 59)(22 95 143 60 119)(23 61 96 120 144)(24 113 62 137 89)(25 47 74 150 112)(26 151 48 105 75)(27 106 152 76 41)(28 77 107 42 145)(29 43 78 146 108)(30 147 44 109 79)(31 110 148 80 45)(32 73 111 46 149)
(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 78 121 18)(2 75 122 23)(3 80 123 20)(4 77 124 17)(5 74 125 22)(6 79 126 19)(7 76 127 24)(8 73 128 21)(9 120 160 48)(10 117 153 45)(11 114 154 42)(12 119 155 47)(13 116 156 44)(14 113 157 41)(15 118 158 46)(16 115 159 43)(25 49 60 40)(26 54 61 37)(27 51 62 34)(28 56 63 39)(29 53 64 36)(30 50 57 33)(31 55 58 38)(32 52 59 35)(65 146 102 91)(66 151 103 96)(67 148 104 93)(68 145 97 90)(69 150 98 95)(70 147 99 92)(71 152 100 89)(72 149 101 94)(81 106 130 137)(82 111 131 142)(83 108 132 139)(84 105 133 144)(85 110 134 141)(86 107 135 138)(87 112 136 143)(88 109 129 140)
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 17)(9 44)(10 45)(11 46)(12 47)(13 48)(14 41)(15 42)(16 43)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 61)(34 62)(35 63)(36 64)(37 57)(38 58)(39 59)(40 60)(65 91)(66 92)(67 93)(68 94)(69 95)(70 96)(71 89)(72 90)(73 124)(74 125)(75 126)(76 127)(77 128)(78 121)(79 122)(80 123)(81 137)(82 138)(83 139)(84 140)(85 141)(86 142)(87 143)(88 144)(97 149)(98 150)(99 151)(100 152)(101 145)(102 146)(103 147)(104 148)(105 129)(106 130)(107 131)(108 132)(109 133)(110 134)(111 135)(112 136)(113 157)(114 158)(115 159)(116 160)(117 153)(118 154)(119 155)(120 156)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H4I 5 8A···8H8I8J8K8L10A10B10C10D10E10F10G20A20B
order122222244444444458···88888101010101010102020
size11112244555510102020410···1020202020444888888

38 irreducible representations

dim11111111112222444488
type+++++++++++--+
imageC1C2C2C2C2C2C2C4C4C4D4C4○D4M4(2)C8○D4F5C2×F5C2×F5C22.F5D4.F5D4×F5
kernelC5⋊C87D4C10.C42Dic5⋊C8C23.2F5D4×Dic5C22×C5⋊C8C2×C22.F5C4⋊Dic5C23.D5D4×C10C5⋊C8Dic5C2×C10C10C2×D4C2×C4C23C22C2C2
# reps11121112422244112411

Matrix representation of C5⋊C87D4 in GL6(𝔽41)

100000
010000
00354000
00364000
000007
0000356
,
3200000
0320000
000010
000001
0002200
0013000
,
010000
4000000
001000
000100
0000400
0000040
,
010000
100000
001000
000100
000010
000001

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,35,36,0,0,0,0,40,40,0,0,0,0,0,0,0,35,0,0,0,0,7,6],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,13,0,0,0,0,22,0,0,0,1,0,0,0,0,0,0,1,0,0],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C5⋊C87D4 in GAP, Magma, Sage, TeX

C_5\rtimes C_8\rtimes_7D_4
% in TeX

G:=Group("C5:C8:7D4");
// GroupNames label

G:=SmallGroup(320,1111);
// by ID

G=gap.SmallGroup(320,1111);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,219,136,6278,1595]);
// Polycyclic

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

׿
×
𝔽