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G = S3×C3⋊C8order 144 = 24·32

Direct product of S3 and C3⋊C8

direct product, metabelian, supersoluble, monomial, A-group

Aliases: S3×C3⋊C8, C12.28D6, D6.2Dic3, Dic3.2Dic3, (C3×S3)⋊C8, C33(S3×C8), C4.13S32, C323(C2×C8), (S3×C6).1C4, (C4×S3).3S3, C6.16(C4×S3), (S3×C12).4C2, C324C86C2, C2.1(S3×Dic3), C6.1(C2×Dic3), (C3×Dic3).2C4, (C3×C12).27C22, C31(C2×C3⋊C8), (C3×C3⋊C8)⋊5C2, (C3×C6).8(C2×C4), SmallGroup(144,52)

Series: Derived Chief Lower central Upper central

C1C32 — S3×C3⋊C8
C1C3C32C3×C6C3×C12S3×C12 — S3×C3⋊C8
C32 — S3×C3⋊C8
C1C4

Generators and relations for S3×C3⋊C8
 G = < a,b,c,d | a3=b2=c3=d8=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

3C2
3C2
2C3
3C4
3C22
2C6
3C6
3C6
3C2×C4
3C8
9C8
2C12
3C12
3C2×C6
9C2×C8
3C24
3C3⋊C8
3C2×C12
3C3⋊C8
6C3⋊C8
3S3×C8
3C2×C3⋊C8

Smallest permutation representation of S3×C3⋊C8
On 48 points
Generators in S48
(1 30 35)(2 31 36)(3 32 37)(4 25 38)(5 26 39)(6 27 40)(7 28 33)(8 29 34)(9 20 41)(10 21 42)(11 22 43)(12 23 44)(13 24 45)(14 17 46)(15 18 47)(16 19 48)
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 9)(17 39)(18 40)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 45)(26 46)(27 47)(28 48)(29 41)(30 42)(31 43)(32 44)
(1 35 30)(2 31 36)(3 37 32)(4 25 38)(5 39 26)(6 27 40)(7 33 28)(8 29 34)(9 41 20)(10 21 42)(11 43 22)(12 23 44)(13 45 24)(14 17 46)(15 47 18)(16 19 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)

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

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

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

S3×C3⋊C8 is a maximal subgroup of
S32×C8  C24.64D6  C24.D6  D12.2Dic3  D12.Dic3  D12.22D6  Dic6.20D6  D12.12D6  D12.13D6  C32⋊C6⋊C8  C12.89S32  C12.93S32
S3×C3⋊C8 is a maximal quotient of
C24.61D6  C12.77D12  C12.81D12  C32⋊C6⋊C8  C12.93S32

36 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D6A6B6C6D6E8A8B8C8D8E8F8G8H12A12B12C12D12E12F12G12H24A24B24C24D
order122233344446666688888888121212121212121224242424
size113322411332246633339999222244666666

36 irreducible representations

dim111111122222222444
type++++++-+-+-
imageC1C2C2C2C4C4C8S3S3Dic3D6Dic3C3⋊C8C4×S3S3×C8S32S3×Dic3S3×C3⋊C8
kernelS3×C3⋊C8C3×C3⋊C8C324C8S3×C12C3×Dic3S3×C6C3×S3C3⋊C8C4×S3Dic3C12D6S3C6C3C4C2C1
# reps111122811121424112

Matrix representation of S3×C3⋊C8 in GL4(𝔽5) generated by

1200
1300
0033
0041
,
0010
0001
1000
0100
,
1200
1300
0012
0013
,
0003
0010
0300
1000
G:=sub<GL(4,GF(5))| [1,1,0,0,2,3,0,0,0,0,3,4,0,0,3,1],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[1,1,0,0,2,3,0,0,0,0,1,1,0,0,2,3],[0,0,0,1,0,0,3,0,0,1,0,0,3,0,0,0] >;

S3×C3⋊C8 in GAP, Magma, Sage, TeX

S_3\times C_3\rtimes C_8
% in TeX

G:=Group("S3xC3:C8");
// GroupNames label

G:=SmallGroup(144,52);
// by ID

G=gap.SmallGroup(144,52);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,31,50,490,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^3=d^8=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Subgroup lattice of S3×C3⋊C8 in TeX

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