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G = C5×S3×D8order 480 = 25·3·5

Direct product of C5, S3 and D8

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

Aliases: C5×S3×D8, C4026D6, D244C10, C12027C22, C60.217C23, C32(C10×D8), C84(S3×C10), C1515(C2×D8), (S3×C8)⋊1C10, C242(C2×C10), D4⋊S31C10, (S3×D4)⋊1C10, (C5×D4)⋊16D6, D41(S3×C10), (C3×D8)⋊2C10, (C15×D8)⋊9C2, (S3×C40)⋊10C2, D121(C2×C10), (C5×D24)⋊12C2, D6.12(C5×D4), C6.27(D4×C10), (S3×C10).48D4, C10.181(S3×D4), C30.363(C2×D4), Dic3.3(C5×D4), (C5×D12)⋊18C22, (D4×C15)⋊18C22, C12.1(C22×C10), (C5×Dic3).30D4, (S3×C20).56C22, C20.190(C22×S3), (C5×S3×D4)⋊8C2, C3⋊C85(C2×C10), C4.1(S3×C2×C10), C2.15(C5×S3×D4), (C5×D4⋊S3)⋊9C2, (C3×D4)⋊1(C2×C10), (C5×C3⋊C8)⋊38C22, (C4×S3).7(C2×C10), SmallGroup(480,789)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C5×S3×D8
 G = < a,b,c,d,e | a5=b3=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 484 in 152 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2 [×6], C3, C4, C4, C22 [×9], C5, S3 [×2], S3 [×2], C6, C6 [×2], C8, C8, C2×C4, D4 [×2], D4 [×4], C23 [×2], C10, C10 [×6], Dic3, C12, D6, D6 [×6], C2×C6 [×2], C15, C2×C8, D8, D8 [×3], C2×D4 [×2], C20, C20, C2×C10 [×9], C3⋊C8, C24, C4×S3, D12 [×2], C3⋊D4 [×2], C3×D4 [×2], C22×S3 [×2], C5×S3 [×2], C5×S3 [×2], C30, C30 [×2], C2×D8, C40, C40, C2×C20, C5×D4 [×2], C5×D4 [×4], C22×C10 [×2], S3×C8, D24, D4⋊S3 [×2], C3×D8, S3×D4 [×2], C5×Dic3, C60, S3×C10, S3×C10 [×6], C2×C30 [×2], C2×C40, C5×D8, C5×D8 [×3], D4×C10 [×2], S3×D8, C5×C3⋊C8, C120, S3×C20, C5×D12 [×2], C5×C3⋊D4 [×2], D4×C15 [×2], S3×C2×C10 [×2], C10×D8, S3×C40, C5×D24, C5×D4⋊S3 [×2], C15×D8, C5×S3×D4 [×2], C5×S3×D8
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], C23, C10 [×7], D6 [×3], D8 [×2], C2×D4, C2×C10 [×7], C22×S3, C5×S3, C2×D8, C5×D4 [×2], C22×C10, S3×D4, S3×C10 [×3], C5×D8 [×2], D4×C10, S3×D8, S3×C2×C10, C10×D8, C5×S3×D4, C5×S3×D8

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

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

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

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

105 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B5A5B5C5D6A6B6C8A8B8C8D10A10B10C10D10E···10L10M···10T10U···10AB 12 15A15B15C15D20A20B20C20D20E20F20G20H24A24B30A30B30C30D30E···30L40A···40H40I···40P60A60B60C60D120A···120H
order12222222344555566688881010101010···1010···1010···101215151515202020202020202024243030303030···3040···4040···4060606060120···120
size11334412122261111288226611113···34···412···1242222222266664422228···82···26···644444···4

105 irreducible representations

dim1111111111112222222222224444
type++++++++++++++
imageC1C2C2C2C2C2C5C10C10C10C10C10S3D4D4D6D6D8C5×S3C5×D4C5×D4S3×C10S3×C10C5×D8S3×D4S3×D8C5×S3×D4C5×S3×D8
kernelC5×S3×D8S3×C40C5×D24C5×D4⋊S3C15×D8C5×S3×D4S3×D8S3×C8D24D4⋊S3C3×D8S3×D4C5×D8C5×Dic3S3×C10C40C5×D4C5×S3D8Dic3D6C8D4S3C10C5C2C1
# reps11121244484811112444448161248

Matrix representation of C5×S3×D8 in GL4(𝔽241) generated by

98000
09800
00910
00091
,
024000
124000
0010
0001
,
124000
024000
0010
0001
,
240000
024000
0023011
00230230
,
240000
024000
0011230
00230230
G:=sub<GL(4,GF(241))| [98,0,0,0,0,98,0,0,0,0,91,0,0,0,0,91],[0,1,0,0,240,240,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,240,240,0,0,0,0,1,0,0,0,0,1],[240,0,0,0,0,240,0,0,0,0,230,230,0,0,11,230],[240,0,0,0,0,240,0,0,0,0,11,230,0,0,230,230] >;

C5×S3×D8 in GAP, Magma, Sage, TeX

C_5\times S_3\times D_8
% in TeX

G:=Group("C5xS3xD8");
// GroupNames label

G:=SmallGroup(480,789);
// by ID

G=gap.SmallGroup(480,789);
# by ID

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

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

׿
×
𝔽