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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 [×6], C3, C4, C4, C22 [×2], C22 [×7], C5, S3 [×2], S3 [×2], C6, C6 [×2], C2×C4, D4, D4 [×3], C23 [×2], C10, C10 [×6], Dic3, C12, D6, D6 [×2], D6 [×4], C2×C6 [×2], C15, C2×D4, C20, C20, C2×C10 [×2], C2×C10 [×7], C4×S3, D12, C3⋊D4 [×2], C3×D4, C22×S3 [×2], C5×S3 [×2], C5×S3 [×2], C30, C30 [×2], C2×C20, C5×D4, C5×D4 [×3], C22×C10 [×2], S3×D4, C5×Dic3, C60, S3×C10, S3×C10 [×2], S3×C10 [×4], C2×C30 [×2], D4×C10, S3×C20, C5×D12, C5×C3⋊D4 [×2], D4×C15, S3×C2×C10 [×2], C5×S3×D4
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], C23, C10 [×7], D6 [×3], C2×D4, C2×C10 [×7], C22×S3, C5×S3, C5×D4 [×2], C22×C10, S3×D4, S3×C10 [×3], 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 45 33)(2 41 34)(3 42 35)(4 43 31)(5 44 32)(6 17 15)(7 18 11)(8 19 12)(9 20 13)(10 16 14)(21 58 51)(22 59 52)(23 60 53)(24 56 54)(25 57 55)(26 47 40)(27 48 36)(28 49 37)(29 50 38)(30 46 39)
(11 18)(12 19)(13 20)(14 16)(15 17)(21 51)(22 52)(23 53)(24 54)(25 55)(31 43)(32 44)(33 45)(34 41)(35 42)(36 48)(37 49)(38 50)(39 46)(40 47)
(1 6 26 56)(2 7 27 57)(3 8 28 58)(4 9 29 59)(5 10 30 60)(11 36 25 34)(12 37 21 35)(13 38 22 31)(14 39 23 32)(15 40 24 33)(16 46 53 44)(17 47 54 45)(18 48 55 41)(19 49 51 42)(20 50 52 43)
(6 56)(7 57)(8 58)(9 59)(10 60)(11 25)(12 21)(13 22)(14 23)(15 24)(16 53)(17 54)(18 55)(19 51)(20 52)

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

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

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

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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