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G = C3×D42S3order 144 = 24·32

Direct product of C3 and D42S3

direct product, metabelian, supersoluble, monomial

Aliases: C3×D42S3, Dic63C6, C12.37D6, C62.13C22, (C4×S3)⋊2C6, D42(C3×S3), (C3×D4)⋊5S3, (C3×D4)⋊3C6, C3⋊D42C6, C4.5(S3×C6), (C2×C6).7D6, (S3×C12)⋊6C2, C12.5(C2×C6), D6.2(C2×C6), (C2×Dic3)⋊3C6, (C3×Dic6)⋊8C2, (C6×Dic3)⋊9C2, (D4×C32)⋊4C2, C329(C4○D4), C6.6(C22×C6), C22.1(S3×C6), (C3×C6).24C23, C6.45(C22×S3), Dic3.3(C2×C6), (S3×C6).11C22, (C3×C12).21C22, (C3×Dic3).13C22, (C2×C6).(C2×C6), C2.7(S3×C2×C6), C32(C3×C4○D4), (C3×C3⋊D4)⋊6C2, SmallGroup(144,163)

Series: Derived Chief Lower central Upper central

C1C6 — C3×D42S3
C1C3C6C3×C6S3×C6S3×C12 — C3×D42S3
C3C6 — C3×D42S3
C1C6C3×D4

Generators and relations for C3×D42S3
 G = < a,b,c,d,e | a3=b4=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >

Subgroups: 164 in 88 conjugacy classes, 46 normal (26 characteristic)
C1, C2, C2 [×3], C3 [×2], C3, C4, C4 [×3], C22 [×2], C22, S3, C6 [×2], C6 [×8], C2×C4 [×3], D4, D4 [×2], Q8, C32, Dic3, Dic3 [×2], C12 [×2], C12 [×4], D6, C2×C6 [×4], C2×C6 [×3], C4○D4, C3×S3, C3×C6, C3×C6 [×2], Dic6, C4×S3, C2×Dic3 [×2], C3⋊D4 [×2], C2×C12 [×3], C3×D4 [×2], C3×D4 [×3], C3×Q8, C3×Dic3, C3×Dic3 [×2], C3×C12, S3×C6, C62 [×2], D42S3, C3×C4○D4, C3×Dic6, S3×C12, C6×Dic3 [×2], C3×C3⋊D4 [×2], D4×C32, C3×D42S3
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], C23, D6 [×3], C2×C6 [×7], C4○D4, C3×S3, C22×S3, C22×C6, S3×C6 [×3], D42S3, C3×C4○D4, S3×C2×C6, C3×D42S3

Permutation representations of C3×D42S3
On 24 points - transitive group 24T209
Generators in S24
(1 17 16)(2 18 13)(3 19 14)(4 20 15)(5 23 10)(6 24 11)(7 21 12)(8 22 9)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(2 4)(6 8)(9 11)(13 15)(18 20)(22 24)
(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)

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

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

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

G:=TransitiveGroup(24,209);

C3×D42S3 is a maximal subgroup of
Dic63D6  Dic6.19D6  D12.22D6  Dic6.20D6  Dic6.24D6  Dic612D6  D1213D6  C3×S3×C4○D4  C62.13D6  Dic182C6  C62.16D6
C3×D42S3 is a maximal quotient of
C3×D4×Dic3  C62.13D6  Dic182C6

45 conjugacy classes

class 1 2A2B2C2D3A3B3C3D3E4A4B4C4D4E6A6B6C···6I6J···6O6P6Q12A12B12C12D12E12F12G12H12I12J12K12L12M
order122223333344444666···66···66612121212121212121212121212
size112261122223366112···24···4662233334446666

45 irreducible representations

dim1111111111112222222244
type+++++++++-
imageC1C2C2C2C2C2C3C6C6C6C6C6S3D6D6C4○D4C3×S3S3×C6S3×C6C3×C4○D4D42S3C3×D42S3
kernelC3×D42S3C3×Dic6S3×C12C6×Dic3C3×C3⋊D4D4×C32D42S3Dic6C4×S3C2×Dic3C3⋊D4C3×D4C3×D4C12C2×C6C32D4C4C22C3C3C1
# reps1112212224421122224412

Matrix representation of C3×D42S3 in GL4(𝔽7) generated by

4000
0400
0040
0004
,
0064
3063
4426
6145
,
6160
6623
6533
0436
,
0046
2310
6122
4450
,
3350
6046
6132
0001
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[0,3,4,6,0,0,4,1,6,6,2,4,4,3,6,5],[6,6,6,0,1,6,5,4,6,2,3,3,0,3,3,6],[0,2,6,4,0,3,1,4,4,1,2,5,6,0,2,0],[3,6,6,0,3,0,1,0,5,4,3,0,0,6,2,1] >;

C3×D42S3 in GAP, Magma, Sage, TeX

C_3\times D_4\rtimes_2S_3
% in TeX

G:=Group("C3xD4:2S3");
// GroupNames label

G:=SmallGroup(144,163);
// by ID

G=gap.SmallGroup(144,163);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-3,151,506,260,3461]);
// Polycyclic

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

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