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G = C5×D8⋊S3order 480 = 25·3·5

Direct product of C5 and D8⋊S3

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C5×D8⋊S3, C4020D6, C12031C22, C60.218C23, C82(S3×C10), (C5×D8)⋊6S3, D82(C5×S3), C244(C2×C10), D4⋊S32C10, (C5×D4)⋊17D6, D42(S3×C10), (C3×D8)⋊4C10, (S3×D4)⋊2C10, D6.6(C5×D4), C24⋊C23C10, C8⋊S33C10, (C15×D8)⋊12C2, D4.S31C10, C6.28(D4×C10), D42S31C10, C1530(C8⋊C22), Dic61(C2×C10), D12.1(C2×C10), (S3×C10).42D4, C10.182(S3×D4), C30.364(C2×D4), Dic3.8(C5×D4), (D4×C15)⋊19C22, C12.2(C22×C10), (C5×Dic3).45D4, (S3×C20).36C22, C20.191(C22×S3), (C5×Dic6)⋊16C22, (C5×D12).30C22, (C5×S3×D4)⋊9C2, C3⋊C81(C2×C10), C32(C5×C8⋊C22), C4.2(S3×C2×C10), C2.16(C5×S3×D4), (C5×D4⋊S3)⋊10C2, (C3×D4)⋊2(C2×C10), (C5×C3⋊C8)⋊23C22, (C5×D4.S3)⋊9C2, (C5×C24⋊C2)⋊11C2, (C5×C8⋊S3)⋊11C2, (C5×D42S3)⋊8C2, (C4×S3).1(C2×C10), SmallGroup(480,790)

Series: Derived Chief Lower central Upper central

C1C12 — C5×D8⋊S3
C1C3C6C12C60S3×C20C5×S3×D4 — C5×D8⋊S3
C3C6C12 — C5×D8⋊S3
C1C10C20C5×D8

Generators and relations for C5×D8⋊S3
 G = < a,b,c,d,e | a5=b8=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe=b5, cd=dc, ce=ec, ede=d-1 >

Subgroups: 388 in 136 conjugacy classes, 54 normal (all characteristic)
C1, C2, C2 [×4], C3, C4, C4 [×2], C22 [×6], C5, S3 [×2], C6, C6 [×2], C8, C8, C2×C4 [×2], D4 [×2], D4 [×3], Q8, C23, C10, C10 [×4], Dic3, Dic3, C12, D6, D6 [×3], C2×C6 [×2], C15, M4(2), D8, D8, SD16 [×2], C2×D4, C4○D4, C20, C20 [×2], C2×C10 [×6], C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4 [×2], C3×D4 [×2], C22×S3, C5×S3 [×2], C30, C30 [×2], C8⋊C22, C40, C40, C2×C20 [×2], C5×D4 [×2], C5×D4 [×3], C5×Q8, C22×C10, C8⋊S3, C24⋊C2, D4⋊S3, D4.S3, C3×D8, S3×D4, D42S3, C5×Dic3, C5×Dic3, C60, S3×C10, S3×C10 [×3], C2×C30 [×2], C5×M4(2), C5×D8, C5×D8, C5×SD16 [×2], D4×C10, C5×C4○D4, D8⋊S3, C5×C3⋊C8, C120, C5×Dic6, S3×C20, C5×D12, C10×Dic3, C5×C3⋊D4 [×2], D4×C15 [×2], S3×C2×C10, C5×C8⋊C22, C5×C8⋊S3, C5×C24⋊C2, C5×D4⋊S3, C5×D4.S3, C15×D8, C5×S3×D4, C5×D42S3, C5×D8⋊S3
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], C23, C10 [×7], D6 [×3], C2×D4, C2×C10 [×7], C22×S3, C5×S3, C8⋊C22, C5×D4 [×2], C22×C10, S3×D4, S3×C10 [×3], D4×C10, D8⋊S3, S3×C2×C10, C5×C8⋊C22, C5×S3×D4, C5×D8⋊S3

Smallest permutation representation of C5×D8⋊S3
On 120 points
Generators in S120
(1 62 31 52 24)(2 63 32 53 17)(3 64 25 54 18)(4 57 26 55 19)(5 58 27 56 20)(6 59 28 49 21)(7 60 29 50 22)(8 61 30 51 23)(9 85 110 35 45)(10 86 111 36 46)(11 87 112 37 47)(12 88 105 38 48)(13 81 106 39 41)(14 82 107 40 42)(15 83 108 33 43)(16 84 109 34 44)(65 93 118 77 102)(66 94 119 78 103)(67 95 120 79 104)(68 96 113 80 97)(69 89 114 73 98)(70 90 115 74 99)(71 91 116 75 100)(72 92 117 76 101)
(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)
(1 3)(4 8)(5 7)(9 11)(12 16)(13 15)(18 24)(19 23)(20 22)(25 31)(26 30)(27 29)(33 39)(34 38)(35 37)(41 43)(44 48)(45 47)(50 56)(51 55)(52 54)(57 61)(58 60)(62 64)(65 69)(66 68)(70 72)(73 77)(74 76)(78 80)(81 83)(84 88)(85 87)(89 93)(90 92)(94 96)(97 103)(98 102)(99 101)(105 109)(106 108)(110 112)(113 119)(114 118)(115 117)
(1 99 35)(2 100 36)(3 101 37)(4 102 38)(5 103 39)(6 104 40)(7 97 33)(8 98 34)(9 31 90)(10 32 91)(11 25 92)(12 26 93)(13 27 94)(14 28 95)(15 29 96)(16 30 89)(17 75 111)(18 76 112)(19 77 105)(20 78 106)(21 79 107)(22 80 108)(23 73 109)(24 74 110)(41 58 66)(42 59 67)(43 60 68)(44 61 69)(45 62 70)(46 63 71)(47 64 72)(48 57 65)(49 120 82)(50 113 83)(51 114 84)(52 115 85)(53 116 86)(54 117 87)(55 118 88)(56 119 81)
(2 6)(4 8)(9 90)(10 95)(11 92)(12 89)(13 94)(14 91)(15 96)(16 93)(17 21)(19 23)(26 30)(28 32)(33 97)(34 102)(35 99)(36 104)(37 101)(38 98)(39 103)(40 100)(41 66)(42 71)(43 68)(44 65)(45 70)(46 67)(47 72)(48 69)(49 53)(51 55)(57 61)(59 63)(73 105)(74 110)(75 107)(76 112)(77 109)(78 106)(79 111)(80 108)(81 119)(82 116)(83 113)(84 118)(85 115)(86 120)(87 117)(88 114)

G:=sub<Sym(120)| (1,62,31,52,24)(2,63,32,53,17)(3,64,25,54,18)(4,57,26,55,19)(5,58,27,56,20)(6,59,28,49,21)(7,60,29,50,22)(8,61,30,51,23)(9,85,110,35,45)(10,86,111,36,46)(11,87,112,37,47)(12,88,105,38,48)(13,81,106,39,41)(14,82,107,40,42)(15,83,108,33,43)(16,84,109,34,44)(65,93,118,77,102)(66,94,119,78,103)(67,95,120,79,104)(68,96,113,80,97)(69,89,114,73,98)(70,90,115,74,99)(71,91,116,75,100)(72,92,117,76,101), (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), (1,3)(4,8)(5,7)(9,11)(12,16)(13,15)(18,24)(19,23)(20,22)(25,31)(26,30)(27,29)(33,39)(34,38)(35,37)(41,43)(44,48)(45,47)(50,56)(51,55)(52,54)(57,61)(58,60)(62,64)(65,69)(66,68)(70,72)(73,77)(74,76)(78,80)(81,83)(84,88)(85,87)(89,93)(90,92)(94,96)(97,103)(98,102)(99,101)(105,109)(106,108)(110,112)(113,119)(114,118)(115,117), (1,99,35)(2,100,36)(3,101,37)(4,102,38)(5,103,39)(6,104,40)(7,97,33)(8,98,34)(9,31,90)(10,32,91)(11,25,92)(12,26,93)(13,27,94)(14,28,95)(15,29,96)(16,30,89)(17,75,111)(18,76,112)(19,77,105)(20,78,106)(21,79,107)(22,80,108)(23,73,109)(24,74,110)(41,58,66)(42,59,67)(43,60,68)(44,61,69)(45,62,70)(46,63,71)(47,64,72)(48,57,65)(49,120,82)(50,113,83)(51,114,84)(52,115,85)(53,116,86)(54,117,87)(55,118,88)(56,119,81), (2,6)(4,8)(9,90)(10,95)(11,92)(12,89)(13,94)(14,91)(15,96)(16,93)(17,21)(19,23)(26,30)(28,32)(33,97)(34,102)(35,99)(36,104)(37,101)(38,98)(39,103)(40,100)(41,66)(42,71)(43,68)(44,65)(45,70)(46,67)(47,72)(48,69)(49,53)(51,55)(57,61)(59,63)(73,105)(74,110)(75,107)(76,112)(77,109)(78,106)(79,111)(80,108)(81,119)(82,116)(83,113)(84,118)(85,115)(86,120)(87,117)(88,114)>;

G:=Group( (1,62,31,52,24)(2,63,32,53,17)(3,64,25,54,18)(4,57,26,55,19)(5,58,27,56,20)(6,59,28,49,21)(7,60,29,50,22)(8,61,30,51,23)(9,85,110,35,45)(10,86,111,36,46)(11,87,112,37,47)(12,88,105,38,48)(13,81,106,39,41)(14,82,107,40,42)(15,83,108,33,43)(16,84,109,34,44)(65,93,118,77,102)(66,94,119,78,103)(67,95,120,79,104)(68,96,113,80,97)(69,89,114,73,98)(70,90,115,74,99)(71,91,116,75,100)(72,92,117,76,101), (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), (1,3)(4,8)(5,7)(9,11)(12,16)(13,15)(18,24)(19,23)(20,22)(25,31)(26,30)(27,29)(33,39)(34,38)(35,37)(41,43)(44,48)(45,47)(50,56)(51,55)(52,54)(57,61)(58,60)(62,64)(65,69)(66,68)(70,72)(73,77)(74,76)(78,80)(81,83)(84,88)(85,87)(89,93)(90,92)(94,96)(97,103)(98,102)(99,101)(105,109)(106,108)(110,112)(113,119)(114,118)(115,117), (1,99,35)(2,100,36)(3,101,37)(4,102,38)(5,103,39)(6,104,40)(7,97,33)(8,98,34)(9,31,90)(10,32,91)(11,25,92)(12,26,93)(13,27,94)(14,28,95)(15,29,96)(16,30,89)(17,75,111)(18,76,112)(19,77,105)(20,78,106)(21,79,107)(22,80,108)(23,73,109)(24,74,110)(41,58,66)(42,59,67)(43,60,68)(44,61,69)(45,62,70)(46,63,71)(47,64,72)(48,57,65)(49,120,82)(50,113,83)(51,114,84)(52,115,85)(53,116,86)(54,117,87)(55,118,88)(56,119,81), (2,6)(4,8)(9,90)(10,95)(11,92)(12,89)(13,94)(14,91)(15,96)(16,93)(17,21)(19,23)(26,30)(28,32)(33,97)(34,102)(35,99)(36,104)(37,101)(38,98)(39,103)(40,100)(41,66)(42,71)(43,68)(44,65)(45,70)(46,67)(47,72)(48,69)(49,53)(51,55)(57,61)(59,63)(73,105)(74,110)(75,107)(76,112)(77,109)(78,106)(79,111)(80,108)(81,119)(82,116)(83,113)(84,118)(85,115)(86,120)(87,117)(88,114) );

G=PermutationGroup([(1,62,31,52,24),(2,63,32,53,17),(3,64,25,54,18),(4,57,26,55,19),(5,58,27,56,20),(6,59,28,49,21),(7,60,29,50,22),(8,61,30,51,23),(9,85,110,35,45),(10,86,111,36,46),(11,87,112,37,47),(12,88,105,38,48),(13,81,106,39,41),(14,82,107,40,42),(15,83,108,33,43),(16,84,109,34,44),(65,93,118,77,102),(66,94,119,78,103),(67,95,120,79,104),(68,96,113,80,97),(69,89,114,73,98),(70,90,115,74,99),(71,91,116,75,100),(72,92,117,76,101)], [(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)], [(1,3),(4,8),(5,7),(9,11),(12,16),(13,15),(18,24),(19,23),(20,22),(25,31),(26,30),(27,29),(33,39),(34,38),(35,37),(41,43),(44,48),(45,47),(50,56),(51,55),(52,54),(57,61),(58,60),(62,64),(65,69),(66,68),(70,72),(73,77),(74,76),(78,80),(81,83),(84,88),(85,87),(89,93),(90,92),(94,96),(97,103),(98,102),(99,101),(105,109),(106,108),(110,112),(113,119),(114,118),(115,117)], [(1,99,35),(2,100,36),(3,101,37),(4,102,38),(5,103,39),(6,104,40),(7,97,33),(8,98,34),(9,31,90),(10,32,91),(11,25,92),(12,26,93),(13,27,94),(14,28,95),(15,29,96),(16,30,89),(17,75,111),(18,76,112),(19,77,105),(20,78,106),(21,79,107),(22,80,108),(23,73,109),(24,74,110),(41,58,66),(42,59,67),(43,60,68),(44,61,69),(45,62,70),(46,63,71),(47,64,72),(48,57,65),(49,120,82),(50,113,83),(51,114,84),(52,115,85),(53,116,86),(54,117,87),(55,118,88),(56,119,81)], [(2,6),(4,8),(9,90),(10,95),(11,92),(12,89),(13,94),(14,91),(15,96),(16,93),(17,21),(19,23),(26,30),(28,32),(33,97),(34,102),(35,99),(36,104),(37,101),(38,98),(39,103),(40,100),(41,66),(42,71),(43,68),(44,65),(45,70),(46,67),(47,72),(48,69),(49,53),(51,55),(57,61),(59,63),(73,105),(74,110),(75,107),(76,112),(77,109),(78,106),(79,111),(80,108),(81,119),(82,116),(83,113),(84,118),(85,115),(86,120),(87,117),(88,114)])

90 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C5A5B5C5D6A6B6C8A8B10A10B10C10D10E···10L10M10N10O10P10Q10R10S10T 12 15A15B15C15D20A20B20C20D20E20F20G20H20I20J20K20L24A24B30A30B30C30D30E···30L40A40B40C40D40E40F40G40H60A60B60C60D120A···120H
order12222234445555666881010101010···101010101010101010121515151520202020202020202020202024243030303030···30404040404040404060606060120···120
size114461222612111128841211114···46666121212124222222226666121212124422228···844441212121244444···4

90 irreducible representations

dim11111111111111112222222222444444
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C5C10C10C10C10C10C10C10S3D4D4D6D6C5×S3C5×D4C5×D4S3×C10S3×C10C8⋊C22S3×D4D8⋊S3C5×C8⋊C22C5×S3×D4C5×D8⋊S3
kernelC5×D8⋊S3C5×C8⋊S3C5×C24⋊C2C5×D4⋊S3C5×D4.S3C15×D8C5×S3×D4C5×D42S3D8⋊S3C8⋊S3C24⋊C2D4⋊S3D4.S3C3×D8S3×D4D42S3C5×D8C5×Dic3S3×C10C40C5×D4D8Dic3D6C8D4C15C10C5C3C2C1
# reps11111111444444441111244448112448

Matrix representation of C5×D8⋊S3 in GL6(𝔽241)

20500000
02050000
001000
000100
000010
000001
,
2402390000
110000
004714719494
009419414747
004714747147
009419494194
,
24000000
110000
000010
000001
001000
000100
,
100000
010000
00240100
00240000
00002401
00002400
,
24000000
02400000
00124000
00024000
00001240
00000240

G:=sub<GL(6,GF(241))| [205,0,0,0,0,0,0,205,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[240,1,0,0,0,0,239,1,0,0,0,0,0,0,47,94,47,94,0,0,147,194,147,194,0,0,194,147,47,94,0,0,94,47,147,194],[240,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,240,0,0,0,0,1,0,0,0,0,0,0,0,240,240,0,0,0,0,1,0],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,240,240,0,0,0,0,0,0,1,0,0,0,0,0,240,240] >;

C5×D8⋊S3 in GAP, Magma, Sage, TeX

C_5\times D_8\rtimes S_3
% in TeX

G:=Group("C5xD8:S3");
// GroupNames label

G:=SmallGroup(480,790);
// by ID

G=gap.SmallGroup(480,790);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1766,471,2111,1068,102,15686]);
// Polycyclic

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

׿
×
𝔽