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G = C5⋊(C8⋊D4)  order 320 = 26·5

The semidirect product of C5 and C8⋊D4 acting via C8⋊D4/Q8⋊C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C52C83D4, C52(C8⋊D4), C4⋊C4.28D10, C4.167(D4×D5), D103Q82C2, Q8⋊C418D5, C4⋊D20.4C2, C20.125(C2×D4), (C2×C8).177D10, (C2×Q8).18D10, D205C425C2, C20.20(C4○D4), C4.33(C4○D20), C10.D810C2, (C2×Dic5).41D4, (C22×D5).27D4, C22.202(D4×D5), C10.24(C4⋊D4), C2.18(D40⋊C2), C10.64(C8⋊C22), (C2×C40).201C22, (C2×C20).252C23, (C2×D20).70C22, C4⋊Dic5.96C22, (Q8×C10).35C22, C2.27(D10⋊D4), C2.16(Q16⋊D5), C10.62(C8.C22), (C2×Q8⋊D5)⋊5C2, (C2×C8⋊D5)⋊20C2, (C2×C4×D5).29C22, (C5×Q8⋊C4)⋊23C2, (C2×C10).265(C2×D4), (C5×C4⋊C4).53C22, (C2×C52C8).43C22, (C2×C4).359(C22×D5), SmallGroup(320,439)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C5⋊(C8⋊D4)
C1C5C10C20C2×C20C2×C4×D5C4⋊D20 — C5⋊(C8⋊D4)
C5C10C2×C20 — C5⋊(C8⋊D4)
C1C22C2×C4Q8⋊C4

Generators and relations for C5⋊(C8⋊D4)
 G = < a,b,c,d | a5=b8=c4=d2=1, bab-1=dad=a-1, ac=ca, cbc-1=b-1, dbd=b3, dcd=c-1 >

Subgroups: 582 in 120 conjugacy classes, 39 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×3], C2×C4, C2×C4 [×6], D4 [×4], Q8 [×2], C23 [×2], D5 [×2], C10 [×3], C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C2×C8, C2×C8, M4(2) [×2], SD16 [×2], C22×C4, C2×D4 [×2], C2×Q8, Dic5 [×2], C20 [×2], C20 [×2], D10 [×6], C2×C10, D4⋊C4, Q8⋊C4, C2.D8, C4⋊D4, C22⋊Q8, C2×M4(2), C2×SD16, C52C8 [×2], C40, C4×D5 [×2], D20 [×4], C2×Dic5, C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5, C22×D5, C8⋊D4, C8⋊D5 [×2], C2×C52C8, C10.D4, C4⋊Dic5, D10⋊C4 [×2], Q8⋊D5 [×2], C5×C4⋊C4, C2×C40, C2×C4×D5, C2×D20, C2×D20, Q8×C10, C10.D8, D205C4, C5×Q8⋊C4, C4⋊D20, C2×C8⋊D5, C2×Q8⋊D5, D103Q8, C5⋊(C8⋊D4)
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C8⋊C22, C8.C22, C22×D5, C8⋊D4, C4○D20, D4×D5 [×2], D10⋊D4, D40⋊C2, Q16⋊D5, C5⋊(C8⋊D4)

Smallest permutation representation of C5⋊(C8⋊D4)
On 160 points
Generators in S160
(1 151 103 21 107)(2 108 22 104 152)(3 145 97 23 109)(4 110 24 98 146)(5 147 99 17 111)(6 112 18 100 148)(7 149 101 19 105)(8 106 20 102 150)(9 143 59 51 91)(10 92 52 60 144)(11 137 61 53 93)(12 94 54 62 138)(13 139 63 55 95)(14 96 56 64 140)(15 141 57 49 89)(16 90 50 58 142)(25 36 115 67 128)(26 121 68 116 37)(27 38 117 69 122)(28 123 70 118 39)(29 40 119 71 124)(30 125 72 120 33)(31 34 113 65 126)(32 127 66 114 35)(41 155 136 77 88)(42 81 78 129 156)(43 157 130 79 82)(44 83 80 131 158)(45 159 132 73 84)(46 85 74 133 160)(47 153 134 75 86)(48 87 76 135 154)
(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 41 141 116)(2 48 142 115)(3 47 143 114)(4 46 144 113)(5 45 137 120)(6 44 138 119)(7 43 139 118)(8 42 140 117)(9 66 109 86)(10 65 110 85)(11 72 111 84)(12 71 112 83)(13 70 105 82)(14 69 106 81)(15 68 107 88)(16 67 108 87)(17 73 93 125)(18 80 94 124)(19 79 95 123)(20 78 96 122)(21 77 89 121)(22 76 90 128)(23 75 91 127)(24 74 92 126)(25 104 135 50)(26 103 136 49)(27 102 129 56)(28 101 130 55)(29 100 131 54)(30 99 132 53)(31 98 133 52)(32 97 134 51)(33 147 159 61)(34 146 160 60)(35 145 153 59)(36 152 154 58)(37 151 155 57)(38 150 156 64)(39 149 157 63)(40 148 158 62)
(1 141)(2 144)(3 139)(4 142)(5 137)(6 140)(7 143)(8 138)(9 149)(10 152)(11 147)(12 150)(13 145)(14 148)(15 151)(16 146)(17 53)(18 56)(19 51)(20 54)(21 49)(22 52)(23 55)(24 50)(25 126)(26 121)(27 124)(28 127)(29 122)(30 125)(31 128)(32 123)(33 72)(34 67)(35 70)(36 65)(37 68)(38 71)(39 66)(40 69)(42 44)(43 47)(46 48)(57 107)(58 110)(59 105)(60 108)(61 111)(62 106)(63 109)(64 112)(73 132)(74 135)(75 130)(76 133)(77 136)(78 131)(79 134)(80 129)(81 158)(82 153)(83 156)(84 159)(85 154)(86 157)(87 160)(88 155)(89 103)(90 98)(91 101)(92 104)(93 99)(94 102)(95 97)(96 100)(113 115)(114 118)(117 119)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12222244444455888810···102020202020···2040···40
size1111204022882040224420202···244448···84···4

44 irreducible representations

dim11111111222222222444444
type++++++++++++++++-+++
imageC1C2C2C2C2C2C2C2D4D4D4D5C4○D4D10D10D10C4○D20C8⋊C22C8.C22D4×D5D4×D5D40⋊C2Q16⋊D5
kernelC5⋊(C8⋊D4)C10.D8D205C4C5×Q8⋊C4C4⋊D20C2×C8⋊D5C2×Q8⋊D5D103Q8C52C8C2×Dic5C22×D5Q8⋊C4C20C4⋊C4C2×C8C2×Q8C4C10C10C4C22C2C2
# reps11111111211222228112244

Matrix representation of C5⋊(C8⋊D4) in GL6(𝔽41)

100000
010000
000100
0040600
000001
0000406
,
4000000
0400000
0020361036
0033211431
003152036
0027103321
,
26370000
36150000
00160352
00016396
00352250
00396025
,
4000000
2810000
001000
0064000
0000400
0000351

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,6,0,0,0,0,0,0,0,40,0,0,0,0,1,6],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,20,33,31,27,0,0,36,21,5,10,0,0,10,14,20,33,0,0,36,31,36,21],[26,36,0,0,0,0,37,15,0,0,0,0,0,0,16,0,35,39,0,0,0,16,2,6,0,0,35,39,25,0,0,0,2,6,0,25],[40,28,0,0,0,0,0,1,0,0,0,0,0,0,1,6,0,0,0,0,0,40,0,0,0,0,0,0,40,35,0,0,0,0,0,1] >;

C5⋊(C8⋊D4) in GAP, Magma, Sage, TeX

C_5\rtimes (C_8\rtimes D_4)
% in TeX

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

G:=SmallGroup(320,439);
// by ID

G=gap.SmallGroup(320,439);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,1094,135,184,570,297,136,12550]);
// Polycyclic

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

׿
×
𝔽