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G = C3×C4.6S4order 288 = 25·32

Direct product of C3 and C4.6S4

direct product, non-abelian, soluble

Aliases: C3×C4.6S4, C12.20S4, GL2(𝔽3)⋊3C6, CSU2(𝔽3)⋊3C6, C4.A42C6, C2.9(C6×S4), C4.6(C3×S4), C6.46(C2×S4), Q8.4(S3×C6), (C3×Q8).22D6, (C3×GL2(𝔽3))⋊7C2, (C3×CSU2(𝔽3))⋊7C2, SL2(𝔽3).4(C2×C6), (C3×SL2(𝔽3)).16C22, (C3×C4○D4)⋊3S3, C4○D41(C3×S3), (C3×C4.A4)⋊7C2, SmallGroup(288,903)

Series: Derived Chief Lower central Upper central

C1C2Q8SL2(𝔽3) — C3×C4.6S4
C1C2Q8SL2(𝔽3)C3×SL2(𝔽3)C3×GL2(𝔽3) — C3×C4.6S4
SL2(𝔽3) — C3×C4.6S4
C1C12

Generators and relations for C3×C4.6S4
 G = < a,b,c,d,e,f | a3=b4=e3=f2=1, c2=d2=b2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd-1=b2c, ece-1=b2cd, fcf=cd, ede-1=c, fdf=b2d, fef=e-1 >

Subgroups: 286 in 83 conjugacy classes, 20 normal (all characteristic)
C1, C2, C2 [×2], C3, C3 [×2], C4, C4 [×2], C22 [×2], S3 [×2], C6, C6 [×4], C8 [×2], C2×C4 [×2], D4 [×3], Q8, Q8, C32, Dic3, C12, C12 [×4], D6, C2×C6 [×2], C2×C8, D8, SD16 [×2], Q16, C4○D4, C4○D4, C3×S3 [×2], C3×C6, C24 [×2], SL2(𝔽3), SL2(𝔽3), C4×S3, C2×C12 [×2], C3×D4 [×3], C3×Q8, C3×Q8, C4○D8, C3×Dic3, C3×C12, S3×C6, C2×C24, C3×D8, C3×SD16 [×2], C3×Q16, CSU2(𝔽3), GL2(𝔽3), C4.A4, C4.A4, C3×C4○D4, C3×C4○D4, C3×SL2(𝔽3), S3×C12, C3×C4○D8, C4.6S4, C3×CSU2(𝔽3), C3×GL2(𝔽3), C3×C4.A4, C3×C4.6S4
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D6, C2×C6, C3×S3, S4, S3×C6, C2×S4, C3×S4, C4.6S4, C6×S4, C3×C4.6S4

Smallest permutation representation of C3×C4.6S4
On 48 points
Generators in S48
(1 31 35)(2 32 36)(3 29 33)(4 30 34)(5 14 10)(6 15 11)(7 16 12)(8 13 9)(17 27 21)(18 28 22)(19 25 23)(20 26 24)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(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)
(1 37 3 39)(2 38 4 40)(5 28 7 26)(6 25 8 27)(9 17 11 19)(10 18 12 20)(13 21 15 23)(14 22 16 24)(29 47 31 45)(30 48 32 46)(33 43 35 41)(34 44 36 42)
(1 17 3 19)(2 18 4 20)(5 48 7 46)(6 45 8 47)(9 39 11 37)(10 40 12 38)(13 43 15 41)(14 44 16 42)(21 33 23 35)(22 34 24 36)(25 31 27 29)(26 32 28 30)
(5 48 26)(6 45 27)(7 46 28)(8 47 25)(9 39 19)(10 40 20)(11 37 17)(12 38 18)(13 43 23)(14 44 24)(15 41 21)(16 42 22)
(1 3)(2 4)(5 48)(6 45)(7 46)(8 47)(9 39)(10 40)(11 37)(12 38)(13 43)(14 44)(15 41)(16 42)(29 31)(30 32)(33 35)(34 36)

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

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

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

48 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D6A6B6C6D6E6F6G6H6I8A8B8C8D12A12B12C12D12E12F12G···12L12M12N24A···24H
order1222333334444666666666888812121212121212···12121224···24
size1161211888116121166888121266661111668···812126···6

48 irreducible representations

dim11111111222222333344
type++++++++
imageC1C2C2C2C3C6C6C6S3D6C3×S3S3×C6C4.6S4C3×C4.6S4S4C2×S4C3×S4C6×S4C4.6S4C3×C4.6S4
kernelC3×C4.6S4C3×CSU2(𝔽3)C3×GL2(𝔽3)C3×C4.A4C4.6S4CSU2(𝔽3)GL2(𝔽3)C4.A4C3×C4○D4C3×Q8C4○D4Q8C3C1C12C6C4C2C3C1
# reps11112222112248224424

Matrix representation of C3×C4.6S4 in GL2(𝔽73) generated by

80
08
,
460
046
,
1629
4457
,
2829
5645
,
072
172
,
721
01
G:=sub<GL(2,GF(73))| [8,0,0,8],[46,0,0,46],[16,44,29,57],[28,56,29,45],[0,1,72,72],[72,0,1,1] >;

C3×C4.6S4 in GAP, Magma, Sage, TeX

C_3\times C_4._6S_4
% in TeX

G:=Group("C3xC4.6S4");
// GroupNames label

G:=SmallGroup(288,903);
// by ID

G=gap.SmallGroup(288,903);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,1016,675,2524,655,172,1517,404,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^3=b^4=e^3=f^2=1,c^2=d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d^-1=b^2*c,e*c*e^-1=b^2*c*d,f*c*f=c*d,e*d*e^-1=c,f*d*f=b^2*d,f*e*f=e^-1>;
// generators/relations

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