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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), C23.11(C2×F5), C23.D5.6C4, Dic5⋊C83C2, C10.15(C8○D4), (D4×Dic5).10C2, Dic5.79(C2×D4), 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), (C2×C10).76(C22×C4), (C22×C10).24(C2×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

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

Generators and relations
 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 >

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

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

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

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

Matrix representation G ⊆ 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] >;

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

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

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