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G = C3×S3×M4(2)  order 288 = 25·32

Direct product of C3, S3 and M4(2)

direct product, metabelian, supersoluble, monomial

Aliases: C3×S3×M4(2), C2422D6, C86(S3×C6), (S3×C8)⋊7C6, C247(C2×C6), C8⋊S35C6, (S3×C24)⋊16C2, (C4×S3).1C12, (S3×C12).4C4, C4.15(S3×C12), D6.9(C2×C12), C32(C6×M4(2)), C4.Dic35C6, C12.106(C4×S3), C12.12(C2×C12), (C3×C24)⋊22C22, (C2×C12).321D6, (C3×M4(2))⋊5C6, C22.7(S3×C12), C62.59(C2×C4), (C22×S3).4C12, C6.15(C22×C12), C12.38(C22×C6), Dic3.7(C2×C12), (C6×Dic3).14C4, (C2×Dic3).6C12, C3212(C2×M4(2)), (S3×C12).62C22, (C6×C12).113C22, (C3×C12).170C23, C12.226(C22×S3), (C32×M4(2))⋊7C2, C3⋊C811(C2×C6), (S3×C2×C6).9C4, (S3×C2×C4).3C6, C4.38(S3×C2×C6), C2.16(S3×C2×C12), C6.114(S3×C2×C4), (C3×C3⋊C8)⋊42C22, (S3×C2×C12).12C2, (C2×C4).45(S3×C6), (C2×C6).62(C4×S3), (C2×C6).5(C2×C12), (C3×C8⋊S3)⋊13C2, (S3×C6).24(C2×C4), (C4×S3).16(C2×C6), (C3×C12).67(C2×C4), (C2×C12).24(C2×C6), (C3×C4.Dic3)⋊21C2, (C3×C6).86(C22×C4), (C3×Dic3).32(C2×C4), SmallGroup(288,677)

Series: Derived Chief Lower central Upper central

C1C6 — C3×S3×M4(2)
C1C3C6C12C3×C12S3×C12S3×C2×C12 — C3×S3×M4(2)
C3C6 — C3×S3×M4(2)
C1C12C3×M4(2)

Generators and relations for C3×S3×M4(2)
 G = < a,b,c,d,e | a3=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=d5 >

Subgroups: 282 in 146 conjugacy classes, 78 normal (54 characteristic)
C1, C2, C2 [×4], C3 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], S3 [×2], S3, C6 [×2], C6 [×7], C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], C23, C32, Dic3 [×2], C12 [×4], C12 [×4], D6 [×2], D6 [×2], C2×C6 [×2], C2×C6 [×5], C2×C8 [×2], M4(2), M4(2) [×3], C22×C4, C3×S3 [×2], C3×S3, C3×C6, C3×C6, C3⋊C8 [×2], C24 [×4], C24 [×4], C4×S3 [×4], C2×Dic3, C2×C12 [×2], C2×C12 [×6], C22×S3, C22×C6, C2×M4(2), C3×Dic3 [×2], C3×C12 [×2], S3×C6 [×2], S3×C6 [×2], C62, S3×C8 [×2], C8⋊S3 [×2], C4.Dic3, C2×C24 [×2], C3×M4(2) [×2], C3×M4(2) [×4], S3×C2×C4, C22×C12, C3×C3⋊C8 [×2], C3×C24 [×2], S3×C12 [×4], C6×Dic3, C6×C12, S3×C2×C6, S3×M4(2), C6×M4(2), S3×C24 [×2], C3×C8⋊S3 [×2], C3×C4.Dic3, C32×M4(2), S3×C2×C12, C3×S3×M4(2)
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], S3, C6 [×7], C2×C4 [×6], C23, C12 [×4], D6 [×3], C2×C6 [×7], M4(2) [×2], C22×C4, C3×S3, C4×S3 [×2], C2×C12 [×6], C22×S3, C22×C6, C2×M4(2), S3×C6 [×3], C3×M4(2) [×2], S3×C2×C4, C22×C12, S3×C12 [×2], S3×C2×C6, S3×M4(2), C6×M4(2), S3×C2×C12, C3×S3×M4(2)

Smallest permutation representation of C3×S3×M4(2)
On 48 points
Generators in S48
(1 30 39)(2 31 40)(3 32 33)(4 25 34)(5 26 35)(6 27 36)(7 28 37)(8 29 38)(9 42 18)(10 43 19)(11 44 20)(12 45 21)(13 46 22)(14 47 23)(15 48 24)(16 41 17)
(1 30 39)(2 31 40)(3 32 33)(4 25 34)(5 26 35)(6 27 36)(7 28 37)(8 29 38)(9 18 42)(10 19 43)(11 20 44)(12 21 45)(13 22 46)(14 23 47)(15 24 48)(16 17 41)
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)(17 34)(18 35)(19 36)(20 37)(21 38)(22 39)(23 40)(24 33)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)
(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)
(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)(34 38)(36 40)(41 45)(43 47)

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

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

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

90 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D3E4A4B4C4D4E4F6A6B6C···6G6H6I6J6K6L6M6N6O6P8A8B8C8D8E8F8G8H12A12B12C12D12E···12L12M12N12O12P12Q12R12S12T12U24A···24H24I···24T24U···24AB
order12222233333444444666···6666666666888888881212121212···1212121212121212121224···2424···2424···24
size11233611222112336112···23333444662222666611112···23333444662···24···46···6

90 irreducible representations

dim11111111111111111122222222222244
type+++++++++
imageC1C2C2C2C2C2C3C4C4C4C6C6C6C6C6C12C12C12S3D6D6M4(2)C3×S3C4×S3C4×S3S3×C6S3×C6C3×M4(2)S3×C12S3×C12S3×M4(2)C3×S3×M4(2)
kernelC3×S3×M4(2)S3×C24C3×C8⋊S3C3×C4.Dic3C32×M4(2)S3×C2×C12S3×M4(2)S3×C12C6×Dic3S3×C2×C6S3×C8C8⋊S3C4.Dic3C3×M4(2)S3×C2×C4C4×S3C2×Dic3C22×S3C3×M4(2)C24C2×C12C3×S3M4(2)C12C2×C6C8C2×C4S3C4C22C3C1
# reps12211124224422284412142224284424

Matrix representation of C3×S3×M4(2) in GL4(𝔽73) generated by

1000
0100
0080
0008
,
1000
0100
0080
00064
,
72000
07200
0001
0010
,
0100
46000
00460
00046
,
1000
07200
0010
0001
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,64],[72,0,0,0,0,72,0,0,0,0,0,1,0,0,1,0],[0,46,0,0,1,0,0,0,0,0,46,0,0,0,0,46],[1,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1] >;

C3×S3×M4(2) in GAP, Magma, Sage, TeX

C_3\times S_3\times M_4(2)
% in TeX

G:=Group("C3xS3xM4(2)");
// GroupNames label

G:=SmallGroup(288,677);
// by ID

G=gap.SmallGroup(288,677);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,555,142,102,9414]);
// Polycyclic

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

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