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## G = C3×D6.4D6order 432 = 24·33

### Direct product of C3 and D6.4D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C3×D6.4D6
 Chief series C1 — C3 — C32 — C3×C6 — C32×C6 — S3×C3×C6 — C3×S3×Dic3 — C3×D6.4D6
 Lower central C32 — C3×C6 — C3×D6.4D6
 Upper central C1 — C6 — C2×C6

Generators and relations for C3×D6.4D6
G = < a,b,c,d,e | a3=b6=c2=1, d6=e2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe-1=b-1, dcd-1=bc, ece-1=b4c, ede-1=d5 >

Subgroups: 672 in 218 conjugacy classes, 64 normal (24 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, Q8, C32, C32, C32, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C4○D4, C3×S3, C3×C6, C3×C6, C3×C6, Dic6, C4×S3, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C3×Q8, C33, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C62, C62, C62, D42S3, C3×C4○D4, S3×C32, C32×C6, C32×C6, S3×Dic3, D6⋊S3, C322Q8, C3×Dic6, S3×C12, C6×Dic3, C3×C3⋊D4, C3×C3⋊D4, C2×C3⋊Dic3, D4×C32, C32×Dic3, C3×C3⋊Dic3, S3×C3×C6, C3×C62, D6.4D6, C3×D42S3, C3×S3×Dic3, C3×D6⋊S3, C3×C322Q8, C32×C3⋊D4, C6×C3⋊Dic3, C3×D6.4D6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C4○D4, C3×S3, C22×S3, C22×C6, S32, S3×C6, D42S3, C3×C4○D4, C2×S32, S3×C2×C6, C3×S32, D6.4D6, C3×D42S3, S32×C6, C3×D6.4D6

Permutation representations of C3×D6.4D6
On 24 points - transitive group 24T1283
Generators in S24
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)
(1 11 9 7 5 3)(2 4 6 8 10 12)(13 15 17 19 21 23)(14 24 22 20 18 16)
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 24)(14 15)(16 17)(18 19)(20 21)(22 23)
(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 16 7 22)(2 21 8 15)(3 14 9 20)(4 19 10 13)(5 24 11 18)(6 17 12 23)

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

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

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

G:=TransitiveGroup(24,1283);

72 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C ··· 3H 3I 3J 3K 4A 4B 4C 4D 4E 6A 6B 6C ··· 6J 6K ··· 6Y 6Z 6AA 6AB 6AC 6AD ··· 6AI 12A 12B 12C 12D 12E 12F 12G 12H 12I ··· 12N 12O 12P order 1 2 2 2 2 3 3 3 ··· 3 3 3 3 4 4 4 4 4 6 6 6 ··· 6 6 ··· 6 6 6 6 6 6 ··· 6 12 12 12 12 12 12 12 12 12 ··· 12 12 12 size 1 1 2 6 6 1 1 2 ··· 2 4 4 4 6 6 9 9 18 1 1 2 ··· 2 4 ··· 4 6 6 6 6 12 ··· 12 6 6 6 6 9 9 9 9 12 ··· 12 18 18

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 S3 D6 D6 D6 C4○D4 C3×S3 S3×C6 S3×C6 S3×C6 C3×C4○D4 S32 D4⋊2S3 C2×S32 C3×S32 D6.4D6 C3×D4⋊2S3 S32×C6 C3×D6.4D6 kernel C3×D6.4D6 C3×S3×Dic3 C3×D6⋊S3 C3×C32⋊2Q8 C32×C3⋊D4 C6×C3⋊Dic3 D6.4D6 S3×Dic3 D6⋊S3 C32⋊2Q8 C3×C3⋊D4 C2×C3⋊Dic3 C3×C3⋊D4 C3×Dic3 S3×C6 C62 C33 C3⋊D4 Dic3 D6 C2×C6 C32 C2×C6 C32 C6 C22 C3 C3 C2 C1 # reps 1 2 1 1 2 1 2 4 2 2 4 2 2 2 2 2 2 4 4 4 4 4 1 2 1 2 2 4 2 4

Matrix representation of C3×D6.4D6 in GL4(𝔽7) generated by

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

C3×D6.4D6 in GAP, Magma, Sage, TeX

C_3\times D_6._4D_6
% in TeX

G:=Group("C3xD6.4D6");
// GroupNames label

G:=SmallGroup(432,653);
// by ID

G=gap.SmallGroup(432,653);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,590,303,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^6=c^2=1,d^6=e^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=d*b*d^-1=e*b*e^-1=b^-1,d*c*d^-1=b*c,e*c*e^-1=b^4*c,e*d*e^-1=d^5>;
// generators/relations

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