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## G = C3×D4⋊2S3order 144 = 24·32

### Direct product of C3 and D4⋊2S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C3×D4⋊2S3
 Chief series C1 — C3 — C6 — C3×C6 — S3×C6 — S3×C12 — C3×D4⋊2S3
 Lower central C3 — C6 — C3×D4⋊2S3
 Upper central C1 — C6 — C3×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 2A 2B 2C 2D 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 6A 6B 6C ··· 6I 6J ··· 6O 6P 6Q 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K 12L 12M order 1 2 2 2 2 3 3 3 3 3 4 4 4 4 4 6 6 6 ··· 6 6 ··· 6 6 6 12 12 12 12 12 12 12 12 12 12 12 12 12 size 1 1 2 2 6 1 1 2 2 2 2 3 3 6 6 1 1 2 ··· 2 4 ··· 4 6 6 2 2 3 3 3 3 4 4 4 6 6 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 D6 D6 C4○D4 C3×S3 S3×C6 S3×C6 C3×C4○D4 D4⋊2S3 C3×D4⋊2S3 kernel C3×D4⋊2S3 C3×Dic6 S3×C12 C6×Dic3 C3×C3⋊D4 D4×C32 D4⋊2S3 Dic6 C4×S3 C2×Dic3 C3⋊D4 C3×D4 C3×D4 C12 C2×C6 C32 D4 C4 C22 C3 C3 C1 # reps 1 1 1 2 2 1 2 2 2 4 4 2 1 1 2 2 2 2 4 4 1 2

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

 4 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 0 0 6 4 3 0 6 3 4 4 2 6 6 1 4 5
,
 6 1 6 0 6 6 2 3 6 5 3 3 0 4 3 6
,
 0 0 4 6 2 3 1 0 6 1 2 2 4 4 5 0
,
 3 3 5 0 6 0 4 6 6 1 3 2 0 0 0 1
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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