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G = C3×C4.3S4order 288 = 25·32

Direct product of C3 and C4.3S4

direct product, non-abelian, soluble

Aliases: C3×C4.3S4, C12.17S4, GL2(𝔽3)⋊2C6, C4.A41C6, C4.3(C3×S4), C2.10(C6×S4), C6.47(C2×S4), Q8.5(S3×C6), (C3×Q8).23D6, (C3×GL2(𝔽3))⋊6C2, SL2(𝔽3)⋊2(C2×C6), (C3×SL2(𝔽3))⋊10C22, (C3×C4.A4)⋊6C2, (C3×C4○D4)⋊4S3, C4○D42(C3×S3), SmallGroup(288,904)

Series: Derived Chief Lower central Upper central

C1C2Q8SL2(𝔽3) — C3×C4.3S4
C1C2Q8SL2(𝔽3)C3×SL2(𝔽3)C3×GL2(𝔽3) — C3×C4.3S4
SL2(𝔽3) — C3×C4.3S4
C1C6C12

Generators and relations for C3×C4.3S4
 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, fbf=b-1, dcd-1=b2c, ece-1=b2cd, fcf=cd, ede-1=c, fdf=b2d, fef=e-1 >

Subgroups: 374 in 89 conjugacy classes, 20 normal (16 characteristic)
C1, C2, C2 [×3], C3, C3 [×2], C4, C4, C22 [×5], S3 [×2], C6, C6 [×5], C8 [×2], C2×C4, D4 [×4], Q8, C23, C32, C12, C12 [×3], D6 [×2], C2×C6 [×5], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, C3×S3 [×2], C3×C6, C24 [×2], SL2(𝔽3), SL2(𝔽3), D12, C2×C12, C3×D4 [×4], C3×Q8, C22×C6, C8⋊C22, C3×C12, S3×C6 [×2], C3×M4(2), C3×D8 [×2], C3×SD16 [×2], GL2(𝔽3) [×2], C4.A4, C4.A4, C6×D4, C3×C4○D4, C3×SL2(𝔽3), C3×D12, C3×C8⋊C22, C4.3S4, C3×GL2(𝔽3) [×2], C3×C4.A4, C3×C4.3S4
Quotients: C1, C2 [×3], C3, C22, S3, C6 [×3], D6, C2×C6, C3×S3, S4, S3×C6, C2×S4, C3×S4, C4.3S4, C6×S4, C3×C4.3S4

Smallest permutation representation of C3×C4.3S4
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)
(2 4)(5 48)(6 47)(7 46)(8 45)(9 37)(10 40)(11 39)(12 38)(13 41)(14 44)(15 43)(16 42)(17 19)(21 23)(25 27)(30 32)(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), (2,4)(5,48)(6,47)(7,46)(8,45)(9,37)(10,40)(11,39)(12,38)(13,41)(14,44)(15,43)(16,42)(17,19)(21,23)(25,27)(30,32)(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), (2,4)(5,48)(6,47)(7,46)(8,45)(9,37)(10,40)(11,39)(12,38)(13,41)(14,44)(15,43)(16,42)(17,19)(21,23)(25,27)(30,32)(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)], [(2,4),(5,48),(6,47),(7,46),(8,45),(9,37),(10,40),(11,39),(12,38),(13,41),(14,44),(15,43),(16,42),(17,19),(21,23),(25,27),(30,32),(34,36)])

39 conjugacy classes

class 1 2A2B2C2D3A3B3C3D3E4A4B6A6B6C6D6E6F6G6H6I6J6K8A8B12A12B12C12D12E···12J24A24B24C24D
order12222333334466666666666881212121212···1224242424
size11612121188826116688812121212121222668···812121212

39 irreducible representations

dim1111112222333344
type++++++++
imageC1C2C2C3C6C6S3D6C3×S3S3×C6S4C2×S4C3×S4C6×S4C4.3S4C3×C4.3S4
kernelC3×C4.3S4C3×GL2(𝔽3)C3×C4.A4C4.3S4GL2(𝔽3)C4.A4C3×C4○D4C3×Q8C4○D4Q8C12C6C4C2C3C1
# reps1212421122224436

Matrix representation of C3×C4.3S4 in GL4(𝔽7) generated by

4000
0400
0040
0004
,
2666
0525
6466
6341
,
5164
0223
2266
1441
,
0341
2604
6343
4554
,
5054
6051
6321
4545
,
2666
4301
5613
1341
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[2,0,6,6,6,5,4,3,6,2,6,4,6,5,6,1],[5,0,2,1,1,2,2,4,6,2,6,4,4,3,6,1],[0,2,6,4,3,6,3,5,4,0,4,5,1,4,3,4],[5,6,6,4,0,0,3,5,5,5,2,4,4,1,1,5],[2,4,5,1,6,3,6,3,6,0,1,4,6,1,3,1] >;

C3×C4.3S4 in GAP, Magma, Sage, TeX

C_3\times C_4._3S_4
% in TeX

G:=Group("C3xC4.3S4");
// GroupNames label

G:=SmallGroup(288,904);
// by ID

G=gap.SmallGroup(288,904);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,2,-2,2045,1016,675,2524,655,172,1517,404,285,124]);
// 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,f*b*f=b^-1,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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