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G = C3×C424S3order 288 = 25·32

Direct product of C3 and C424S3

direct product, metabelian, supersoluble, monomial

Aliases: C3×C424S3, C12C2, D121C12, C1229C2, Dic61C12, C12.92D12, C62.101D4, C328C4≀C2, (C4×C12)⋊12S3, (C4×C12)⋊12C6, C4.6(S3×C12), C427(C3×S3), (C3×D12)⋊10C4, C12.84(C4×S3), C4○D12.1C6, C4.17(C3×D12), C12.33(C3×D4), C4.Dic31C6, C12.16(C2×C12), C6.42(D6⋊C4), (C3×Dic6)⋊10C4, (C3×C12).135D4, (C2×C12).430D6, (C6×C12).311C22, C31(C3×C4≀C2), C2.3(C3×D6⋊C4), (C2×C4).69(S3×C6), (C2×C6).36(C3×D4), C6.1(C3×C22⋊C4), (C3×C12).96(C2×C4), (C3×C4○D12).3C2, C22.7(C3×C3⋊D4), (C2×C12).112(C2×C6), (C2×C6).60(C3⋊D4), (C3×C4.Dic3)⋊17C2, (C3×C6).41(C22⋊C4), SmallGroup(288,239)

Series: Derived Chief Lower central Upper central

C1C12 — C3×C424S3
C1C3C6C12C2×C12C6×C12C3×C4○D12 — C3×C424S3
C3C6C12 — C3×C424S3
C1C12C2×C12C4×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 2A2B2C3A3B3C3D3E4A4B4C···4G4H6A6B6C···6M6N6O8A8B12A12B12C12D12E···12AX12AY12AZ24A24B24C24D
order122233333444···44666···666881212121212···12121224242424
size1121211222112···212112···21212121211112···2121212121212

90 irreducible representations

dim111111111111222222222222222222
type+++++++++
imageC1C2C2C2C3C4C4C6C6C6C12C12S3D4D4D6C3×S3C4×S3D12C3×D4C3⋊D4C3×D4C4≀C2S3×C6S3×C12C3×D12C3×C3⋊D4C424S3C3×C4≀C2C3×C424S3
kernelC3×C424S3C3×C4.Dic3C122C3×C4○D12C424S3C3×Dic6C3×D12C4.Dic3C4×C12C4○D12Dic6D12C4×C12C3×C12C62C2×C12C42C12C12C12C2×C6C2×C6C32C2×C4C4C4C22C3C3C1
# reps1111222222441111222222424448816

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

30
03
,
312
93
,
09
100
,
68
66
,
120
01
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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