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G = C3×C42⋊4S3order 288 = 25·32

Direct product of C3 and C42⋊4S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C3×C42⋊4S3
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C6×C12 — C3×C4○D12 — C3×C42⋊4S3
 Lower central C3 — C6 — C12 — C3×C42⋊4S3
 Upper central C1 — C12 — C2×C12 — C4×C12

Generators and relations for C3×C424S3
G = < a,b,c,d,e | a3=b4=c4=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, ebe=bc=cb, bd=db, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 242 in 103 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C4 [×3], C22, C22, S3, C6 [×2], C6 [×6], C8, C2×C4, C2×C4 [×2], D4 [×2], Q8, C32, Dic3, C12 [×4], C12 [×11], D6, C2×C6 [×2], C2×C6 [×2], C42, M4(2), C4○D4, C3×S3, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12 [×2], C2×C12 [×6], C3×D4 [×2], C3×Q8, C4≀C2, C3×Dic3, C3×C12 [×2], C3×C12 [×2], S3×C6, C62, C4.Dic3, C4×C12 [×2], C4×C12, C3×M4(2), C4○D12, C3×C4○D4, C3×C3⋊C8, C3×Dic6, S3×C12, C3×D12, C3×C3⋊D4, C6×C12, C6×C12, C424S3, C3×C4≀C2, C3×C4.Dic3, C122, C3×C4○D12, C3×C424S3
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, D4 [×2], C12 [×2], D6, C2×C6, C22⋊C4, C3×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×D4 [×2], C4≀C2, S3×C6, D6⋊C4, C3×C22⋊C4, S3×C12, C3×D12, C3×C3⋊D4, C424S3, C3×C4≀C2, C3×D6⋊C4, C3×C424S3

Permutation representations of C3×C424S3
On 24 points - transitive group 24T627
Generators in S24
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 17 24)(14 18 21)(15 19 22)(16 20 23)
(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 10 4 7)(2 11 5 8)(3 12 6 9)(13 16 15 14)(17 20 19 18)(21 24 23 22)
(1 3 2)(4 6 5)(7 9 8)(10 12 11)(13 17 24)(14 18 21)(15 19 22)(16 20 23)
(1 20)(2 23)(3 16)(4 18)(5 21)(6 14)(7 19)(8 22)(9 15)(10 17)(11 24)(12 13)

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

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

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

G:=TransitiveGroup(24,627);

90 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C ··· 4G 4H 6A 6B 6C ··· 6M 6N 6O 8A 8B 12A 12B 12C 12D 12E ··· 12AX 12AY 12AZ 24A 24B 24C 24D order 1 2 2 2 3 3 3 3 3 4 4 4 ··· 4 4 6 6 6 ··· 6 6 6 8 8 12 12 12 12 12 ··· 12 12 12 24 24 24 24 size 1 1 2 12 1 1 2 2 2 1 1 2 ··· 2 12 1 1 2 ··· 2 12 12 12 12 1 1 1 1 2 ··· 2 12 12 12 12 12 12

90 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 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C3 C4 C4 C6 C6 C6 C12 C12 S3 D4 D4 D6 C3×S3 C4×S3 D12 C3×D4 C3⋊D4 C3×D4 C4≀C2 S3×C6 S3×C12 C3×D12 C3×C3⋊D4 C42⋊4S3 C3×C4≀C2 C3×C42⋊4S3 kernel C3×C42⋊4S3 C3×C4.Dic3 C122 C3×C4○D12 C42⋊4S3 C3×Dic6 C3×D12 C4.Dic3 C4×C12 C4○D12 Dic6 D12 C4×C12 C3×C12 C62 C2×C12 C42 C12 C12 C12 C2×C6 C2×C6 C32 C2×C4 C4 C4 C22 C3 C3 C1 # reps 1 1 1 1 2 2 2 2 2 2 4 4 1 1 1 1 2 2 2 2 2 2 4 2 4 4 4 8 8 16

Matrix representation of C3×C424S3 in GL2(𝔽13) generated by

 3 0 0 3
,
 3 12 9 3
,
 0 9 10 0
,
 6 8 6 6
,
 12 0 0 1
G:=sub<GL(2,GF(13))| [3,0,0,3],[3,9,12,3],[0,10,9,0],[6,6,8,6],[12,0,0,1] >;

C3×C424S3 in GAP, Magma, Sage, TeX

C_3\times C_4^2\rtimes_4S_3
% in TeX

G:=Group("C3xC4^2:4S3");
// GroupNames label

G:=SmallGroup(288,239);
// by ID

G=gap.SmallGroup(288,239);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,365,92,1683,2524,102,9414]);
// Polycyclic

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

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