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## G = C3×C62⋊C4order 432 = 24·33

### Direct product of C3 and C62⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C3×C62⋊C4
 Chief series C1 — C32 — C3×C6 — C2×C3⋊S3 — C6×C3⋊S3 — C6×C32⋊C4 — C3×C62⋊C4
 Lower central C32 — C3×C6 — C3×C62⋊C4
 Upper central C1 — C6 — C2×C6

Generators and relations for C3×C62⋊C4
G = < a,b,c,d | a3=b6=c6=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c, dcd-1=b4c >

Subgroups: 764 in 152 conjugacy classes, 28 normal (20 characteristic)
C1, C2, C2 [×4], C3, C3 [×4], C4 [×2], C22, C22 [×4], S3 [×8], C6, C6 [×16], C2×C4 [×2], C23, C32, C32 [×4], C12 [×2], D6 [×12], C2×C6, C2×C6 [×8], C22⋊C4, C3×S3 [×8], C3⋊S3 [×2], C3⋊S3, C3×C6, C3×C6 [×13], C2×C12 [×2], C22×S3 [×2], C22×C6, C33, C32⋊C4 [×2], S3×C6 [×12], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C62 [×4], C3×C22⋊C4, C3×C3⋊S3 [×2], C3×C3⋊S3, C32×C6, C32×C6, C2×C32⋊C4 [×2], S3×C2×C6 [×2], C22×C3⋊S3, C3×C32⋊C4 [×2], C6×C3⋊S3 [×2], C6×C3⋊S3 [×2], C3×C62, C62⋊C4, C6×C32⋊C4 [×2], C2×C6×C3⋊S3, C3×C62⋊C4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, D4 [×2], C12 [×2], C2×C6, C22⋊C4, C2×C12, C3×D4 [×2], C32⋊C4, C3×C22⋊C4, C2×C32⋊C4, C3×C32⋊C4, C62⋊C4, C6×C32⋊C4, C3×C62⋊C4

Permutation representations of C3×C62⋊C4
On 24 points - transitive group 24T1288
Generators in S24
(1 10 6)(2 9 5)(3 7 12)(4 8 11)(13 14 15)(16 18 17)(19 21 20)(22 23 24)
(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 5 10 2 6 9)(3 8 12 4 7 11)(13 24 15 23 14 22)(16 20 18 19 17 21)
(1 16 7 22)(2 19 8 15)(3 24 6 17)(4 14 5 20)(9 21 11 13)(10 18 12 23)

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

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

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

G:=TransitiveGroup(24,1288);

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C ··· 3H 4A 4B 4C 4D 6A 6B 6C 6D 6E ··· 6V 6W 6X 6Y 6Z 6AA 6AB 12A ··· 12H order 1 2 2 2 2 2 3 3 3 ··· 3 4 4 4 4 6 6 6 6 6 ··· 6 6 6 6 6 6 6 12 ··· 12 size 1 1 2 9 9 18 1 1 4 ··· 4 18 18 18 18 1 1 2 2 4 ··· 4 9 9 9 9 18 18 18 ··· 18

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 4 4 4 4 4 4 type + + + + + + + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 D4 C3×D4 C32⋊C4 C2×C32⋊C4 C3×C32⋊C4 C62⋊C4 C6×C32⋊C4 C3×C62⋊C4 kernel C3×C62⋊C4 C6×C32⋊C4 C2×C6×C3⋊S3 C62⋊C4 C6×C3⋊S3 C3×C62 C2×C32⋊C4 C22×C3⋊S3 C2×C3⋊S3 C62 C3×C3⋊S3 C3⋊S3 C2×C6 C6 C22 C3 C2 C1 # reps 1 2 1 2 2 2 4 2 4 4 2 4 2 2 4 4 4 8

Matrix representation of C3×C62⋊C4 in GL4(𝔽7) generated by

 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 5 4 6 1 0 4 5 6 3 0 3 4 6 6 6 2
,
 1 3 0 1 5 6 4 6 1 1 3 4 1 5 0 3
,
 5 1 4 1 6 1 2 1 5 0 3 0 1 6 3 5
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[5,0,3,6,4,4,0,6,6,5,3,6,1,6,4,2],[1,5,1,1,3,6,1,5,0,4,3,0,1,6,4,3],[5,6,5,1,1,1,0,6,4,2,3,3,1,1,0,5] >;

C3×C62⋊C4 in GAP, Magma, Sage, TeX

C_3\times C_6^2\rtimes C_4
% in TeX

G:=Group("C3xC6^2:C4");
// GroupNames label

G:=SmallGroup(432,634);
// by ID

G=gap.SmallGroup(432,634);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,3,84,365,14117,362,18822,1203]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^6=c^6=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c,d*c*d^-1=b^4*c>;
// generators/relations

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