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G = C5×S3×D4order 240 = 24·3·5

Direct product of C5, S3 and D4

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

Aliases: C5×S3×D4, C206D6, D123C10, C607C22, C30.52C23, C12⋊(C2×C10), C32(D4×C10), C41(S3×C10), (C2×C10)⋊7D6, C1515(C2×D4), (S3×C20)⋊6C2, (C4×S3)⋊1C10, D62(C2×C10), (D4×C15)⋊8C2, (C3×D4)⋊2C10, (C5×D12)⋊9C2, C3⋊D41C10, (C2×C30)⋊7C22, C222(S3×C10), (C22×S3)⋊2C10, Dic31(C2×C10), C6.5(C22×C10), (S3×C10)⋊10C22, C10.42(C22×S3), (C5×Dic3)⋊8C22, (C2×C6)⋊(C2×C10), (S3×C2×C10)⋊6C2, C2.6(S3×C2×C10), (C5×C3⋊D4)⋊5C2, SmallGroup(240,169)

Series: Derived Chief Lower central Upper central

C1C6 — C5×S3×D4
C1C3C6C30S3×C10S3×C2×C10 — C5×S3×D4
C3C6 — C5×S3×D4
C1C10C5×D4

Generators and relations for C5×S3×D4
 G = < a,b,c,d,e | a5=b3=c2=d4=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: 240 in 108 conjugacy classes, 50 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, S3, C6, C6, C2×C4, D4, D4, C23, C10, C10, Dic3, C12, D6, D6, D6, C2×C6, C15, C2×D4, C20, C20, C2×C10, C2×C10, C4×S3, D12, C3⋊D4, C3×D4, C22×S3, C5×S3, C5×S3, C30, C30, C2×C20, C5×D4, C5×D4, C22×C10, S3×D4, C5×Dic3, C60, S3×C10, S3×C10, S3×C10, C2×C30, D4×C10, S3×C20, C5×D12, C5×C3⋊D4, D4×C15, S3×C2×C10, C5×S3×D4
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C2×C10, C22×S3, C5×S3, C5×D4, C22×C10, S3×D4, S3×C10, D4×C10, S3×C2×C10, C5×S3×D4

Smallest permutation representation of C5×S3×D4
On 60 points
Generators in S60
(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)
(1 39 33)(2 40 34)(3 36 35)(4 37 31)(5 38 32)(6 20 13)(7 16 14)(8 17 15)(9 18 11)(10 19 12)(21 58 51)(22 59 52)(23 60 53)(24 56 54)(25 57 55)(26 41 47)(27 42 48)(28 43 49)(29 44 50)(30 45 46)
(6 13)(7 14)(8 15)(9 11)(10 12)(21 51)(22 52)(23 53)(24 54)(25 55)(31 37)(32 38)(33 39)(34 40)(35 36)(41 47)(42 48)(43 49)(44 50)(45 46)
(1 18 26 56)(2 19 27 57)(3 20 28 58)(4 16 29 59)(5 17 30 60)(6 49 21 35)(7 50 22 31)(8 46 23 32)(9 47 24 33)(10 48 25 34)(11 41 54 39)(12 42 55 40)(13 43 51 36)(14 44 52 37)(15 45 53 38)
(6 21)(7 22)(8 23)(9 24)(10 25)(11 54)(12 55)(13 51)(14 52)(15 53)(16 59)(17 60)(18 56)(19 57)(20 58)

G:=sub<Sym(60)| (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), (1,39,33)(2,40,34)(3,36,35)(4,37,31)(5,38,32)(6,20,13)(7,16,14)(8,17,15)(9,18,11)(10,19,12)(21,58,51)(22,59,52)(23,60,53)(24,56,54)(25,57,55)(26,41,47)(27,42,48)(28,43,49)(29,44,50)(30,45,46), (6,13)(7,14)(8,15)(9,11)(10,12)(21,51)(22,52)(23,53)(24,54)(25,55)(31,37)(32,38)(33,39)(34,40)(35,36)(41,47)(42,48)(43,49)(44,50)(45,46), (1,18,26,56)(2,19,27,57)(3,20,28,58)(4,16,29,59)(5,17,30,60)(6,49,21,35)(7,50,22,31)(8,46,23,32)(9,47,24,33)(10,48,25,34)(11,41,54,39)(12,42,55,40)(13,43,51,36)(14,44,52,37)(15,45,53,38), (6,21)(7,22)(8,23)(9,24)(10,25)(11,54)(12,55)(13,51)(14,52)(15,53)(16,59)(17,60)(18,56)(19,57)(20,58)>;

G:=Group( (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), (1,39,33)(2,40,34)(3,36,35)(4,37,31)(5,38,32)(6,20,13)(7,16,14)(8,17,15)(9,18,11)(10,19,12)(21,58,51)(22,59,52)(23,60,53)(24,56,54)(25,57,55)(26,41,47)(27,42,48)(28,43,49)(29,44,50)(30,45,46), (6,13)(7,14)(8,15)(9,11)(10,12)(21,51)(22,52)(23,53)(24,54)(25,55)(31,37)(32,38)(33,39)(34,40)(35,36)(41,47)(42,48)(43,49)(44,50)(45,46), (1,18,26,56)(2,19,27,57)(3,20,28,58)(4,16,29,59)(5,17,30,60)(6,49,21,35)(7,50,22,31)(8,46,23,32)(9,47,24,33)(10,48,25,34)(11,41,54,39)(12,42,55,40)(13,43,51,36)(14,44,52,37)(15,45,53,38), (6,21)(7,22)(8,23)(9,24)(10,25)(11,54)(12,55)(13,51)(14,52)(15,53)(16,59)(17,60)(18,56)(19,57)(20,58) );

G=PermutationGroup([[(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)], [(1,39,33),(2,40,34),(3,36,35),(4,37,31),(5,38,32),(6,20,13),(7,16,14),(8,17,15),(9,18,11),(10,19,12),(21,58,51),(22,59,52),(23,60,53),(24,56,54),(25,57,55),(26,41,47),(27,42,48),(28,43,49),(29,44,50),(30,45,46)], [(6,13),(7,14),(8,15),(9,11),(10,12),(21,51),(22,52),(23,53),(24,54),(25,55),(31,37),(32,38),(33,39),(34,40),(35,36),(41,47),(42,48),(43,49),(44,50),(45,46)], [(1,18,26,56),(2,19,27,57),(3,20,28,58),(4,16,29,59),(5,17,30,60),(6,49,21,35),(7,50,22,31),(8,46,23,32),(9,47,24,33),(10,48,25,34),(11,41,54,39),(12,42,55,40),(13,43,51,36),(14,44,52,37),(15,45,53,38)], [(6,21),(7,22),(8,23),(9,24),(10,25),(11,54),(12,55),(13,51),(14,52),(15,53),(16,59),(17,60),(18,56),(19,57),(20,58)]])

C5×S3×D4 is a maximal subgroup of   D2010D6  D12.9D10  D2013D6  D1214D10

75 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B5A5B5C5D6A6B6C10A10B10C10D10E···10L10M···10T10U···10AB 12 15A15B15C15D20A20B20C20D20E20F20G20H30A30B30C30D30E···30L60A60B60C60D
order1222222234455556661010101010···1010···1010···10121515151520202020202020203030303030···3060606060
size11223366226111124411112···23···36···6422222222666622224···44444

75 irreducible representations

dim1111111111112222222244
type+++++++++++
imageC1C2C2C2C2C2C5C10C10C10C10C10S3D4D6D6C5×S3C5×D4S3×C10S3×C10S3×D4C5×S3×D4
kernelC5×S3×D4S3×C20C5×D12C5×C3⋊D4D4×C15S3×C2×C10S3×D4C4×S3D12C3⋊D4C3×D4C22×S3C5×D4C5×S3C20C2×C10D4S3C4C22C5C1
# reps1112124448481212484814

Matrix representation of C5×S3×D4 in GL4(𝔽61) generated by

20000
02000
00340
00034
,
06000
16000
0010
0001
,
0100
1000
0010
0001
,
60000
06000
0012
006060
,
1000
0100
0010
006060
G:=sub<GL(4,GF(61))| [20,0,0,0,0,20,0,0,0,0,34,0,0,0,0,34],[0,1,0,0,60,60,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,1,60,0,0,2,60],[1,0,0,0,0,1,0,0,0,0,1,60,0,0,0,60] >;

C5×S3×D4 in GAP, Magma, Sage, TeX

C_5\times S_3\times D_4
% in TeX

G:=Group("C5xS3xD4");
// GroupNames label

G:=SmallGroup(240,169);
// by ID

G=gap.SmallGroup(240,169);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-3,404,5765]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^3=c^2=d^4=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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