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## G = C3×S3×D4order 144 = 24·32

### Direct product of C3, S3 and D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3×S3×D4
 Chief series C1 — C3 — C6 — C3×C6 — S3×C6 — S3×C2×C6 — C3×S3×D4
 Lower central C3 — C6 — C3×S3×D4
 Upper central C1 — C6 — C3×D4

Generators and relations for C3×S3×D4
G = < a,b,c,d,e | a3=b3=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 260 in 116 conjugacy classes, 50 normal (26 characteristic)
C1, C2, C2 [×6], C3 [×2], C3, C4, C4, C22 [×2], C22 [×7], S3 [×2], S3 [×2], C6 [×2], C6 [×11], C2×C4, D4, D4 [×3], C23 [×2], C32, Dic3, C12 [×2], C12 [×2], D6, D6 [×2], D6 [×4], C2×C6 [×4], C2×C6 [×9], C2×D4, C3×S3 [×2], C3×S3 [×2], C3×C6, C3×C6 [×2], C4×S3, D12, C3⋊D4 [×2], C2×C12, C3×D4 [×2], C3×D4 [×4], C22×S3 [×2], C22×C6 [×2], C3×Dic3, C3×C12, S3×C6, S3×C6 [×2], S3×C6 [×4], C62 [×2], S3×D4, C6×D4, S3×C12, C3×D12, C3×C3⋊D4 [×2], D4×C32, S3×C2×C6 [×2], C3×S3×D4
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], C2×D4, C3×S3, C3×D4 [×2], C22×S3, C22×C6, S3×C6 [×3], S3×D4, C6×D4, S3×C2×C6, C3×S3×D4

Permutation representations of C3×S3×D4
On 24 points - transitive group 24T208
Generators in S24
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 10 23)(6 11 24)(7 12 21)(8 9 22)
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 23 10)(6 24 11)(7 21 12)(8 22 9)
(1 22)(2 23)(3 24)(4 21)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)
(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 4)(2 3)(5 6)(7 8)(9 12)(10 11)(13 14)(15 16)(17 20)(18 19)(21 22)(23 24)

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

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

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

G:=TransitiveGroup(24,208);

C3×S3×D4 is a maximal subgroup of   D129D6  D12.7D6  D1212D6  D1213D6

45 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C ··· 6I 6J 6K 6L 6M 6N ··· 6S 6T 6U 6V 6W 12A 12B 12C 12D 12E 12F 12G order 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 6 6 6 ··· 6 6 6 6 6 6 ··· 6 6 6 6 6 12 12 12 12 12 12 12 size 1 1 2 2 3 3 6 6 1 1 2 2 2 2 6 1 1 2 ··· 2 3 3 3 3 4 ··· 4 6 6 6 6 2 2 4 4 4 6 6

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 S3 D4 D6 D6 C3×S3 C3×D4 S3×C6 S3×C6 S3×D4 C3×S3×D4 kernel C3×S3×D4 S3×C12 C3×D12 C3×C3⋊D4 D4×C32 S3×C2×C6 S3×D4 C4×S3 D12 C3⋊D4 C3×D4 C22×S3 C3×D4 C3×S3 C12 C2×C6 D4 S3 C4 C22 C3 C1 # reps 1 1 1 2 1 2 2 2 2 4 2 4 1 2 1 2 2 4 2 4 1 2

Matrix representation of C3×S3×D4 in GL4(𝔽7) generated by

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

C3×S3×D4 in GAP, Magma, Sage, TeX

C_3\times S_3\times D_4
% in TeX

G:=Group("C3xS3xD4");
// GroupNames label

G:=SmallGroup(144,162);
// by ID

G=gap.SmallGroup(144,162);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-3,260,3461]);
// Polycyclic

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

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