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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, C3, C4, C4, C22, C5, S3, S3, C6, C6, C8, C8, C2×C4, D4, D4, C23, C10, C10, Dic3, C12, D6, D6, C2×C6, C15, C2×C8, D8, D8, C2×D4, C20, C20, C2×C10, C3⋊C8, C24, C4×S3, D12, C3⋊D4, C3×D4, C22×S3, C5×S3, C5×S3, C30, C30, C2×D8, C40, C40, C2×C20, C5×D4, C5×D4, C22×C10, S3×C8, D24, D4⋊S3, C3×D8, S3×D4, C5×Dic3, C60, S3×C10, S3×C10, C2×C30, C2×C40, C5×D8, C5×D8, D4×C10, S3×D8, C5×C3⋊C8, C120, S3×C20, C5×D12, C5×C3⋊D4, D4×C15, S3×C2×C10, C10×D8, S3×C40, C5×D24, C5×D4⋊S3, C15×D8, C5×S3×D4, C5×S3×D8
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, D8, C2×D4, C2×C10, C22×S3, C5×S3, C2×D8, C5×D4, C22×C10, S3×D4, S3×C10, C5×D8, 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 90 52 74)(2 63 91 53 75)(3 64 92 54 76)(4 57 93 55 77)(5 58 94 56 78)(6 59 95 49 79)(7 60 96 50 80)(8 61 89 51 73)(9 29 110 22 45)(10 30 111 23 46)(11 31 112 24 47)(12 32 105 17 48)(13 25 106 18 41)(14 26 107 19 42)(15 27 108 20 43)(16 28 109 21 44)(33 119 82 103 66)(34 120 83 104 67)(35 113 84 97 68)(36 114 85 98 69)(37 115 86 99 70)(38 116 87 100 71)(39 117 88 101 72)(40 118 81 102 65)
(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 37 90)(10 38 91)(11 39 92)(12 40 93)(13 33 94)(14 34 95)(15 35 96)(16 36 89)(25 119 56)(26 120 49)(27 113 50)(28 114 51)(29 115 52)(30 116 53)(31 117 54)(32 118 55)(41 66 58)(42 67 59)(43 68 60)(44 69 61)(45 70 62)(46 71 63)(47 72 64)(48 65 57)(73 109 85)(74 110 86)(75 111 87)(76 112 88)(77 105 81)(78 106 82)(79 107 83)(80 108 84)
(9 37)(10 38)(11 39)(12 40)(13 33)(14 34)(15 35)(16 36)(17 102)(18 103)(19 104)(20 97)(21 98)(22 99)(23 100)(24 101)(25 119)(26 120)(27 113)(28 114)(29 115)(30 116)(31 117)(32 118)(41 66)(42 67)(43 68)(44 69)(45 70)(46 71)(47 72)(48 65)(81 105)(82 106)(83 107)(84 108)(85 109)(86 110)(87 111)(88 112)
(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 28)(26 27)(29 32)(30 31)(33 36)(34 35)(37 40)(38 39)(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 86)(82 85)(83 84)(87 88)(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)| (1,62,90,52,74)(2,63,91,53,75)(3,64,92,54,76)(4,57,93,55,77)(5,58,94,56,78)(6,59,95,49,79)(7,60,96,50,80)(8,61,89,51,73)(9,29,110,22,45)(10,30,111,23,46)(11,31,112,24,47)(12,32,105,17,48)(13,25,106,18,41)(14,26,107,19,42)(15,27,108,20,43)(16,28,109,21,44)(33,119,82,103,66)(34,120,83,104,67)(35,113,84,97,68)(36,114,85,98,69)(37,115,86,99,70)(38,116,87,100,71)(39,117,88,101,72)(40,118,81,102,65), (1,22,99)(2,23,100)(3,24,101)(4,17,102)(5,18,103)(6,19,104)(7,20,97)(8,21,98)(9,37,90)(10,38,91)(11,39,92)(12,40,93)(13,33,94)(14,34,95)(15,35,96)(16,36,89)(25,119,56)(26,120,49)(27,113,50)(28,114,51)(29,115,52)(30,116,53)(31,117,54)(32,118,55)(41,66,58)(42,67,59)(43,68,60)(44,69,61)(45,70,62)(46,71,63)(47,72,64)(48,65,57)(73,109,85)(74,110,86)(75,111,87)(76,112,88)(77,105,81)(78,106,82)(79,107,83)(80,108,84), (9,37)(10,38)(11,39)(12,40)(13,33)(14,34)(15,35)(16,36)(17,102)(18,103)(19,104)(20,97)(21,98)(22,99)(23,100)(24,101)(25,119)(26,120)(27,113)(28,114)(29,115)(30,116)(31,117)(32,118)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,65)(81,105)(82,106)(83,107)(84,108)(85,109)(86,110)(87,111)(88,112), (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)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35)(37,40)(38,39)(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)(81,86)(82,85)(83,84)(87,88)(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( (1,62,90,52,74)(2,63,91,53,75)(3,64,92,54,76)(4,57,93,55,77)(5,58,94,56,78)(6,59,95,49,79)(7,60,96,50,80)(8,61,89,51,73)(9,29,110,22,45)(10,30,111,23,46)(11,31,112,24,47)(12,32,105,17,48)(13,25,106,18,41)(14,26,107,19,42)(15,27,108,20,43)(16,28,109,21,44)(33,119,82,103,66)(34,120,83,104,67)(35,113,84,97,68)(36,114,85,98,69)(37,115,86,99,70)(38,116,87,100,71)(39,117,88,101,72)(40,118,81,102,65), (1,22,99)(2,23,100)(3,24,101)(4,17,102)(5,18,103)(6,19,104)(7,20,97)(8,21,98)(9,37,90)(10,38,91)(11,39,92)(12,40,93)(13,33,94)(14,34,95)(15,35,96)(16,36,89)(25,119,56)(26,120,49)(27,113,50)(28,114,51)(29,115,52)(30,116,53)(31,117,54)(32,118,55)(41,66,58)(42,67,59)(43,68,60)(44,69,61)(45,70,62)(46,71,63)(47,72,64)(48,65,57)(73,109,85)(74,110,86)(75,111,87)(76,112,88)(77,105,81)(78,106,82)(79,107,83)(80,108,84), (9,37)(10,38)(11,39)(12,40)(13,33)(14,34)(15,35)(16,36)(17,102)(18,103)(19,104)(20,97)(21,98)(22,99)(23,100)(24,101)(25,119)(26,120)(27,113)(28,114)(29,115)(30,116)(31,117)(32,118)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,65)(81,105)(82,106)(83,107)(84,108)(85,109)(86,110)(87,111)(88,112), (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)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35)(37,40)(38,39)(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)(81,86)(82,85)(83,84)(87,88)(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([[(1,62,90,52,74),(2,63,91,53,75),(3,64,92,54,76),(4,57,93,55,77),(5,58,94,56,78),(6,59,95,49,79),(7,60,96,50,80),(8,61,89,51,73),(9,29,110,22,45),(10,30,111,23,46),(11,31,112,24,47),(12,32,105,17,48),(13,25,106,18,41),(14,26,107,19,42),(15,27,108,20,43),(16,28,109,21,44),(33,119,82,103,66),(34,120,83,104,67),(35,113,84,97,68),(36,114,85,98,69),(37,115,86,99,70),(38,116,87,100,71),(39,117,88,101,72),(40,118,81,102,65)], [(1,22,99),(2,23,100),(3,24,101),(4,17,102),(5,18,103),(6,19,104),(7,20,97),(8,21,98),(9,37,90),(10,38,91),(11,39,92),(12,40,93),(13,33,94),(14,34,95),(15,35,96),(16,36,89),(25,119,56),(26,120,49),(27,113,50),(28,114,51),(29,115,52),(30,116,53),(31,117,54),(32,118,55),(41,66,58),(42,67,59),(43,68,60),(44,69,61),(45,70,62),(46,71,63),(47,72,64),(48,65,57),(73,109,85),(74,110,86),(75,111,87),(76,112,88),(77,105,81),(78,106,82),(79,107,83),(80,108,84)], [(9,37),(10,38),(11,39),(12,40),(13,33),(14,34),(15,35),(16,36),(17,102),(18,103),(19,104),(20,97),(21,98),(22,99),(23,100),(24,101),(25,119),(26,120),(27,113),(28,114),(29,115),(30,116),(31,117),(32,118),(41,66),(42,67),(43,68),(44,69),(45,70),(46,71),(47,72),(48,65),(81,105),(82,106),(83,107),(84,108),(85,109),(86,110),(87,111),(88,112)], [(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),(25,28),(26,27),(29,32),(30,31),(33,36),(34,35),(37,40),(38,39),(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),(81,86),(82,85),(83,84),(87,88),(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

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