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G = S3×C24order 144 = 24·32

Direct product of C24 and S3

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

Aliases: S3×C24, C244C6, D6.2C12, C12.64D6, Dic3.2C12, C3⋊C86C6, (C3×C24)⋊6C2, C31(C2×C24), C326(C2×C8), (S3×C6).4C4, (C4×S3).3C6, C2.1(S3×C12), C4.12(S3×C6), C6.21(C4×S3), C6.1(C2×C12), (S3×C12).6C2, C12.13(C2×C6), (C3×Dic3).4C4, (C3×C12).42C22, (C3×C3⋊C8)⋊13C2, (C3×C6).17(C2×C4), SmallGroup(144,69)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C24
C1C3C6C12C3×C12S3×C12 — S3×C24
C3 — S3×C24
C1C24

Generators and relations for S3×C24
 G = < a,b,c | a24=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C2
2C3
3C4
3C22
2C6
3C6
3C6
3C2×C4
3C8
2C12
3C12
3C2×C6
3C2×C8
2C24
3C2×C12
3C24
3C2×C24

Smallest permutation representation of S3×C24
On 48 points
Generators in S48
(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 9 17)(2 10 18)(3 11 19)(4 12 20)(5 13 21)(6 14 22)(7 15 23)(8 16 24)(25 41 33)(26 42 34)(27 43 35)(28 44 36)(29 45 37)(30 46 38)(31 47 39)(32 48 40)
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)

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

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

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

S3×C24 is a maximal subgroup of   C24.61D6  C24.63D6  C24.64D6  D6.1D12  D247S3  D6.3D12

72 conjugacy classes

class 1 2A2B2C3A3B3C3D3E4A4B4C4D6A6B6C6D6E6F6G6H6I8A8B8C8D8E8F8G8H12A12B12C12D12E···12J12K12L12M12N24A···24H24I···24T24U···24AB
order1222333334444666666666888888881212121212···121212121224···2424···2424···24
size11331122211331122233331111333311112···233331···12···23···3

72 irreducible representations

dim1111111111111122222222
type++++++
imageC1C2C2C2C3C4C4C6C6C6C8C12C12C24S3D6C3×S3C4×S3S3×C6S3×C8S3×C12S3×C24
kernelS3×C24C3×C3⋊C8C3×C24S3×C12S3×C8C3×Dic3S3×C6C3⋊C8C24C4×S3C3×S3Dic3D6S3C24C12C8C6C4C3C2C1
# reps11112222228441611222448

Matrix representation of S3×C24 in GL2(𝔽73) generated by

70
07
,
640
08
,
072
720
G:=sub<GL(2,GF(73))| [7,0,0,7],[64,0,0,8],[0,72,72,0] >;

S3×C24 in GAP, Magma, Sage, TeX

S_3\times C_{24}
% in TeX

G:=Group("S3xC24");
// GroupNames label

G:=SmallGroup(144,69);
// by ID

G=gap.SmallGroup(144,69);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-3,79,69,3461]);
// Polycyclic

G:=Group<a,b,c|a^24=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of S3×C24 in TeX

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