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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 — C3⋊Dic3 — C3×C3⋊Dic3 — C3×C32⋊2C8 — 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=1, d4=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c, dcd-1=b4c >

Subgroups: 332 in 92 conjugacy classes, 24 normal (20 characteristic)
C1, C2, C2, C3, C3 [×4], C4 [×2], C22, C6, C6 [×13], C8 [×2], C2×C4, C32, C32 [×4], Dic3 [×4], C12 [×2], C2×C6, C2×C6 [×4], M4(2), C3×C6, C3×C6 [×13], C24 [×2], C2×Dic3 [×2], C2×C12, C33, C3×Dic3 [×4], C3⋊Dic3 [×2], C62, C62 [×4], C3×M4(2), C32×C6, C32×C6, C322C8 [×2], C6×Dic3 [×2], C2×C3⋊Dic3, C3×C3⋊Dic3 [×2], C3×C62, C62.C4, C3×C322C8 [×2], C6×C3⋊Dic3, C3×C62.C4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, C12 [×2], C2×C6, M4(2), C2×C12, C32⋊C4, C3×M4(2), 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 24T1289
Generators in S24
(1 13 21)(2 14 22)(3 15 23)(4 16 24)(5 9 17)(6 10 18)(7 11 19)(8 12 20)
(1 13 21)(2 10 22 6 14 18)(3 23 15)(4 20 16 8 24 12)(5 9 17)(7 19 11)
(1 17 13 5 21 9)(2 6)(3 11 23 7 15 19)(4 8)(10 14)(12 16)(18 22)(20 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)

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

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

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

G:=TransitiveGroup(24,1289);

54 conjugacy classes

 class 1 2A 2B 3A 3B 3C ··· 3H 4A 4B 4C 6A 6B 6C 6D 6E ··· 6V 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 24A ··· 24H order 1 2 2 3 3 3 ··· 3 4 4 4 6 6 6 6 6 ··· 6 8 8 8 8 12 12 12 12 12 12 24 ··· 24 size 1 1 2 1 1 4 ··· 4 9 9 18 1 1 2 2 4 ··· 4 18 18 18 18 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 M4(2) C3×M4(2) C32⋊C4 C2×C32⋊C4 C3×C32⋊C4 C62.C4 C6×C32⋊C4 C3×C62.C4 kernel C3×C62.C4 C3×C32⋊2C8 C6×C3⋊Dic3 C62.C4 C3×C3⋊Dic3 C3×C62 C32⋊2C8 C2×C3⋊Dic3 C3⋊Dic3 C62 C33 C32 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
,
 2 6 0 4 4 0 5 1 4 4 0 6 0 0 0 5
,
 4 5 3 4 3 2 2 1 4 4 1 6 0 0 0 6
,
 6 1 6 3 6 6 6 3 5 2 1 4 3 3 4 1
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[2,4,4,0,6,0,4,0,0,5,0,0,4,1,6,5],[4,3,4,0,5,2,4,0,3,2,1,0,4,1,6,6],[6,6,5,3,1,6,2,3,6,6,1,4,3,3,4,1] >;

C3×C62.C4 in GAP, Magma, Sage, TeX

C_3\times C_6^2.C_4
% in TeX

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

G:=SmallGroup(432,633);
// by ID

G=gap.SmallGroup(432,633);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,3,84,365,80,14117,362,18822,1203]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^6=c^6=1,d^4=c^3,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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