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## G = S3×C2×C6order 72 = 23·32

### Direct product of C2×C6 and S3

Aliases: S3×C2×C6, C624C2, C322C23, C6⋊(C2×C6), (C2×C6)⋊5C6, C3⋊(C22×C6), (C3×C6)⋊2C22, SmallGroup(72,48)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C2×C6
 Chief series C1 — C3 — C32 — C3×S3 — S3×C6 — S3×C2×C6
 Lower central C3 — S3×C2×C6
 Upper central C1 — C2×C6

Generators and relations for S3×C2×C6
G = < a,b,c,d | a2=b6=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 118 in 69 conjugacy classes, 42 normal (10 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C22, C22 [×6], S3 [×4], C6 [×6], C6 [×7], C23, C32, D6 [×6], C2×C6 [×2], C2×C6 [×7], C3×S3 [×4], C3×C6 [×3], C22×S3, C22×C6, S3×C6 [×6], C62, S3×C2×C6
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], C23, D6 [×3], C2×C6 [×7], C3×S3, C22×S3, C22×C6, S3×C6 [×3], S3×C2×C6

Permutation representations of S3×C2×C6
On 24 points - transitive group 24T68
Generators in S24
(1 8)(2 9)(3 10)(4 11)(5 12)(6 7)(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 3 5)(2 4 6)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 23 21)(20 24 22)
(1 17)(2 18)(3 13)(4 14)(5 15)(6 16)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)

G:=sub<Sym(24)| (1,8)(2,9)(3,10)(4,11)(5,12)(6,7)(13,22)(14,23)(15,24)(16,19)(17,20)(18,21), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,23,21)(20,24,22), (1,17)(2,18)(3,13)(4,14)(5,15)(6,16)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)>;

G:=Group( (1,8)(2,9)(3,10)(4,11)(5,12)(6,7)(13,22)(14,23)(15,24)(16,19)(17,20)(18,21), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,23,21)(20,24,22), (1,17)(2,18)(3,13)(4,14)(5,15)(6,16)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24) );

G=PermutationGroup([(1,8),(2,9),(3,10),(4,11),(5,12),(6,7),(13,22),(14,23),(15,24),(16,19),(17,20),(18,21)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,3,5),(2,4,6),(7,9,11),(8,10,12),(13,17,15),(14,18,16),(19,23,21),(20,24,22)], [(1,17),(2,18),(3,13),(4,14),(5,15),(6,16),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24)])

G:=TransitiveGroup(24,68);

S3×C2×C6 is a maximal subgroup of   D6⋊Dic3

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 6A ··· 6F 6G ··· 6O 6P ··· 6W order 1 2 2 2 2 2 2 2 3 3 3 3 3 6 ··· 6 6 ··· 6 6 ··· 6 size 1 1 1 1 3 3 3 3 1 1 2 2 2 1 ··· 1 2 ··· 2 3 ··· 3

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C3 C6 C6 S3 D6 C3×S3 S3×C6 kernel S3×C2×C6 S3×C6 C62 C22×S3 D6 C2×C6 C2×C6 C6 C22 C2 # reps 1 6 1 2 12 2 1 3 2 6

Matrix representation of S3×C2×C6 in GL3(𝔽7) generated by

 1 0 0 0 6 0 0 0 6
,
 3 0 0 0 2 0 0 0 2
,
 1 0 0 0 4 0 0 0 2
,
 6 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(7))| [1,0,0,0,6,0,0,0,6],[3,0,0,0,2,0,0,0,2],[1,0,0,0,4,0,0,0,2],[6,0,0,0,0,1,0,1,0] >;

S3×C2×C6 in GAP, Magma, Sage, TeX

S_3\times C_2\times C_6
% in TeX

G:=Group("S3xC2xC6");
// GroupNames label

G:=SmallGroup(72,48);
// by ID

G=gap.SmallGroup(72,48);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-3,1204]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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