Copied to
clipboard

G = C5⋊(C82D4)  order 320 = 26·5

The semidirect product of C5 and C82D4 acting via C82D4/D4⋊C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C52C82D4, C51(C82D4), C202D42C2, C4⋊D204C2, C4⋊C4.13D10, C4.161(D4×D5), D4⋊C419D5, (C2×D4).29D10, (C2×C8).170D10, C20.111(C2×D4), C20.Q86C2, D205C420C2, C4.27(C4○D20), C20.10(C4○D4), (C2×Dic5).31D4, (C22×D5).22D4, C22.180(D4×D5), C2.18(D8⋊D5), C10.18(C4⋊D4), C2.11(D40⋊C2), C10.36(C8⋊C22), (C2×C20).222C23, (C2×C40).187C22, (C2×D20).56C22, (D4×C10).43C22, C4⋊Dic5.75C22, C2.21(D10⋊D4), (C2×D4⋊D5)⋊5C2, (C2×C8⋊D5)⋊18C2, (C5×D4⋊C4)⋊25C2, (C2×C4×D5).18C22, (C2×C10).235(C2×D4), (C5×C4⋊C4).23C22, (C2×C52C8).20C22, (C2×C4).329(C22×D5), SmallGroup(320,409)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C5⋊(C82D4)
C1C5C10C20C2×C20C2×C4×D5C4⋊D20 — C5⋊(C82D4)
C5C10C2×C20 — C5⋊(C82D4)
C1C22C2×C4D4⋊C4

Generators and relations for C5⋊(C82D4)
 G = < a,b,c,d | a5=b8=c4=d2=1, bab-1=dad=a-1, ac=ca, cbc-1=b3, dbd=b-1, dcd=c-1 >

Subgroups: 662 in 130 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, C22×C4, C2×D4, C2×D4, Dic5, C20, C20, D10, C2×C10, C2×C10, D4⋊C4, D4⋊C4, C4.Q8, C4⋊D4, C2×M4(2), C2×D8, C52C8, C40, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, C82D4, C8⋊D5, C2×C52C8, C4⋊Dic5, D10⋊C4, D4⋊D5, C23.D5, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×D20, C2×D20, C2×C5⋊D4, D4×C10, C20.Q8, D205C4, C5×D4⋊C4, C4⋊D20, C2×C8⋊D5, C2×D4⋊D5, C202D4, C5⋊(C82D4)
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C8⋊C22, C22×D5, C82D4, C4○D20, D4×D5, D10⋊D4, D8⋊D5, D40⋊C2, C5⋊(C82D4)

Smallest permutation representation of C5⋊(C82D4)
On 160 points
Generators in S160
(1 74 14 143 43)(2 44 144 15 75)(3 76 16 137 45)(4 46 138 9 77)(5 78 10 139 47)(6 48 140 11 79)(7 80 12 141 41)(8 42 142 13 73)(17 108 117 149 85)(18 86 150 118 109)(19 110 119 151 87)(20 88 152 120 111)(21 112 113 145 81)(22 82 146 114 105)(23 106 115 147 83)(24 84 148 116 107)(25 133 155 56 91)(26 92 49 156 134)(27 135 157 50 93)(28 94 51 158 136)(29 129 159 52 95)(30 96 53 160 130)(31 131 153 54 89)(32 90 55 154 132)(33 69 98 62 121)(34 122 63 99 70)(35 71 100 64 123)(36 124 57 101 72)(37 65 102 58 125)(38 126 59 103 66)(39 67 104 60 127)(40 128 61 97 68)
(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)
(1 125 119 55)(2 128 120 50)(3 123 113 53)(4 126 114 56)(5 121 115 51)(6 124 116 54)(7 127 117 49)(8 122 118 52)(9 66 82 133)(10 69 83 136)(11 72 84 131)(12 67 85 134)(13 70 86 129)(14 65 87 132)(15 68 88 135)(16 71 81 130)(17 26 141 104)(18 29 142 99)(19 32 143 102)(20 27 144 97)(21 30 137 100)(22 25 138 103)(23 28 139 98)(24 31 140 101)(33 147 158 78)(34 150 159 73)(35 145 160 76)(36 148 153 79)(37 151 154 74)(38 146 155 77)(39 149 156 80)(40 152 157 75)(41 60 108 92)(42 63 109 95)(43 58 110 90)(44 61 111 93)(45 64 112 96)(46 59 105 91)(47 62 106 94)(48 57 107 89)
(1 115)(2 114)(3 113)(4 120)(5 119)(6 118)(7 117)(8 116)(9 20)(10 19)(11 18)(12 17)(13 24)(14 23)(15 22)(16 21)(25 135)(26 134)(27 133)(28 132)(29 131)(30 130)(31 129)(32 136)(33 58)(34 57)(35 64)(36 63)(37 62)(38 61)(39 60)(40 59)(41 149)(42 148)(43 147)(44 146)(45 145)(46 152)(47 151)(48 150)(50 56)(51 55)(52 54)(65 98)(66 97)(67 104)(68 103)(69 102)(70 101)(71 100)(72 99)(73 107)(74 106)(75 105)(76 112)(77 111)(78 110)(79 109)(80 108)(81 137)(82 144)(83 143)(84 142)(85 141)(86 140)(87 139)(88 138)(89 159)(90 158)(91 157)(92 156)(93 155)(94 154)(95 153)(96 160)(121 125)(122 124)(126 128)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E5A5B8A8B8C8D10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222224444455888810···1010101010202020202020202040···40
size1111820402282040224420202···28888444488884···4

44 irreducible representations

dim1111111122222222244444
type+++++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D5C4○D4D10D10D10C4○D20C8⋊C22D4×D5D4×D5D8⋊D5D40⋊C2
kernelC5⋊(C82D4)C20.Q8D205C4C5×D4⋊C4C4⋊D20C2×C8⋊D5C2×D4⋊D5C202D4C52C8C2×Dic5C22×D5D4⋊C4C20C4⋊C4C2×C8C2×D4C4C10C4C22C2C2
# reps1111111121122222822244

Matrix representation of C5⋊(C82D4) in GL6(𝔽41)

100000
010000
000100
0040600
000001
0000406
,
3750000
1340000
00300629
0016112435
003863629
00293405
,
540000
14360000
00735220
00612022
001835346
006233529
,
4000000
2310000
001000
0064000
0010400
00640351

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],[37,13,0,0,0,0,5,4,0,0,0,0,0,0,30,16,38,29,0,0,0,11,6,3,0,0,6,24,36,40,0,0,29,35,29,5],[5,14,0,0,0,0,4,36,0,0,0,0,0,0,7,6,18,6,0,0,35,12,35,23,0,0,22,0,34,35,0,0,0,22,6,29],[40,23,0,0,0,0,0,1,0,0,0,0,0,0,1,6,1,6,0,0,0,40,0,40,0,0,0,0,40,35,0,0,0,0,0,1] >;

C5⋊(C82D4) in GAP, Magma, Sage, TeX

C_5\rtimes (C_8\rtimes_2D_4)
% in TeX

G:=Group("C5:(C8:2D4)");
// GroupNames label

G:=SmallGroup(320,409);
// by ID

G=gap.SmallGroup(320,409);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,1094,135,570,297,136,12550]);
// 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^3,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽