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## G = S3×C3⋊D4order 144 = 24·32

### Direct product of S3 and C3⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — S3×C3⋊D4
 Chief series C1 — C3 — C32 — C3×C6 — S3×C6 — C2×S32 — S3×C3⋊D4
 Lower central C32 — C3×C6 — S3×C3⋊D4
 Upper central C1 — C2 — C22

Generators and relations for S3×C3⋊D4
G = < a,b,c,d,e | a3=b2=c3=d4=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 412 in 116 conjugacy classes, 36 normal (32 characteristic)
C1, C2, C2 [×6], C3 [×2], C3, C4 [×2], C22, C22 [×8], S3 [×2], S3 [×5], C6 [×2], C6 [×8], C2×C4, D4 [×4], C23 [×2], C32, Dic3, Dic3 [×3], C12, D6 [×3], D6 [×9], C2×C6 [×2], C2×C6 [×6], C2×D4, C3×S3 [×2], C3×S3 [×2], C3⋊S3, C3×C6, C3×C6, C4×S3, D12, C2×Dic3, C3⋊D4, C3⋊D4 [×6], C3×D4, C22×S3, C22×S3 [×2], C22×C6, C3×Dic3, C3⋊Dic3, S32 [×2], S3×C6 [×3], S3×C6 [×2], C2×C3⋊S3, C62, S3×D4, C2×C3⋊D4, S3×Dic3, D6⋊S3, C3⋊D12, C3×C3⋊D4, C327D4, C2×S32, S3×C2×C6, S3×C3⋊D4
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C3⋊D4 [×2], C22×S3 [×2], S32, S3×D4, C2×C3⋊D4, C2×S32, S3×C3⋊D4

Character table of S3×C3⋊D4

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M 12 size 1 1 2 3 3 6 6 18 2 2 4 6 18 2 2 2 2 4 4 4 4 6 6 6 6 12 12 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 -1 -1 1 1 -1 1 1 1 -1 1 -1 1 1 -1 -1 1 -1 -1 -1 1 1 -1 1 -1 linear of order 2 ρ3 1 1 1 1 1 -1 1 -1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 linear of order 2 ρ4 1 1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 1 1 -1 -1 1 -1 -1 -1 1 1 -1 -1 1 linear of order 2 ρ5 1 1 -1 1 1 -1 -1 -1 1 1 1 1 1 -1 1 1 -1 -1 1 -1 -1 1 -1 -1 1 -1 1 linear of order 2 ρ6 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ7 1 1 -1 1 1 1 -1 1 1 1 1 -1 -1 -1 1 1 -1 -1 1 -1 -1 1 -1 -1 1 1 -1 linear of order 2 ρ8 1 1 1 -1 -1 1 -1 -1 1 1 1 1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ9 2 2 2 -2 -2 0 -2 0 -1 2 -1 0 0 -1 -1 2 -1 2 -1 -1 -1 1 1 1 1 0 0 orthogonal lifted from D6 ρ10 2 -2 0 -2 2 0 0 0 2 2 2 0 0 0 -2 -2 0 0 -2 0 0 -2 0 0 2 0 0 orthogonal lifted from D4 ρ11 2 2 -2 -2 -2 0 2 0 -1 2 -1 0 0 1 -1 2 1 -2 -1 1 1 1 -1 -1 1 0 0 orthogonal lifted from D6 ρ12 2 2 2 0 0 2 0 0 2 -1 -1 2 0 2 2 -1 2 -1 -1 -1 -1 0 0 0 0 -1 -1 orthogonal lifted from S3 ρ13 2 2 2 0 0 -2 0 0 2 -1 -1 -2 0 2 2 -1 2 -1 -1 -1 -1 0 0 0 0 1 1 orthogonal lifted from D6 ρ14 2 -2 0 2 -2 0 0 0 2 2 2 0 0 0 -2 -2 0 0 -2 0 0 2 0 0 -2 0 0 orthogonal lifted from D4 ρ15 2 2 -2 2 2 0 -2 0 -1 2 -1 0 0 1 -1 2 1 -2 -1 1 1 -1 1 1 -1 0 0 orthogonal lifted from D6 ρ16 2 2 2 2 2 0 2 0 -1 2 -1 0 0 -1 -1 2 -1 2 -1 -1 -1 -1 -1 -1 -1 0 0 orthogonal lifted from S3 ρ17 2 2 -2 0 0 -2 0 0 2 -1 -1 2 0 -2 2 -1 -2 1 -1 1 1 0 0 0 0 1 -1 orthogonal lifted from D6 ρ18 2 2 -2 0 0 2 0 0 2 -1 -1 -2 0 -2 2 -1 -2 1 -1 1 1 0 0 0 0 -1 1 orthogonal lifted from D6 ρ19 2 -2 0 2 -2 0 0 0 -1 2 -1 0 0 -√-3 1 -2 √-3 0 1 √-3 -√-3 -1 -√-3 √-3 1 0 0 complex lifted from C3⋊D4 ρ20 2 -2 0 2 -2 0 0 0 -1 2 -1 0 0 √-3 1 -2 -√-3 0 1 -√-3 √-3 -1 √-3 -√-3 1 0 0 complex lifted from C3⋊D4 ρ21 2 -2 0 -2 2 0 0 0 -1 2 -1 0 0 -√-3 1 -2 √-3 0 1 √-3 -√-3 1 √-3 -√-3 -1 0 0 complex lifted from C3⋊D4 ρ22 2 -2 0 -2 2 0 0 0 -1 2 -1 0 0 √-3 1 -2 -√-3 0 1 -√-3 √-3 1 -√-3 √-3 -1 0 0 complex lifted from C3⋊D4 ρ23 4 4 4 0 0 0 0 0 -2 -2 1 0 0 -2 -2 -2 -2 -2 1 1 1 0 0 0 0 0 0 orthogonal lifted from S32 ρ24 4 -4 0 0 0 0 0 0 4 -2 -2 0 0 0 -4 2 0 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ25 4 4 -4 0 0 0 0 0 -2 -2 1 0 0 2 -2 -2 2 2 1 -1 -1 0 0 0 0 0 0 orthogonal lifted from C2×S32 ρ26 4 -4 0 0 0 0 0 0 -2 -2 1 0 0 2√-3 2 2 -2√-3 0 -1 √-3 -√-3 0 0 0 0 0 0 complex faithful ρ27 4 -4 0 0 0 0 0 0 -2 -2 1 0 0 -2√-3 2 2 2√-3 0 -1 -√-3 √-3 0 0 0 0 0 0 complex faithful

Permutation representations of S3×C3⋊D4
On 24 points - transitive group 24T204
Generators in S24
(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 24)(2 21)(3 22)(4 23)(5 15)(6 16)(7 13)(8 14)(9 19)(10 20)(11 17)(12 18)
(1 17 16)(2 13 18)(3 19 14)(4 15 20)(5 10 23)(6 24 11)(7 12 21)(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)
(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)(5,23,10)(6,24,11)(7,21,12)(8,22,9), (1,24)(2,21)(3,22)(4,23)(5,15)(6,16)(7,13)(8,14)(9,19)(10,20)(11,17)(12,18), (1,17,16)(2,13,18)(3,19,14)(4,15,20)(5,10,23)(6,24,11)(7,12,21)(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), (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)(5,23,10)(6,24,11)(7,21,12)(8,22,9), (1,24)(2,21)(3,22)(4,23)(5,15)(6,16)(7,13)(8,14)(9,19)(10,20)(11,17)(12,18), (1,17,16)(2,13,18)(3,19,14)(4,15,20)(5,10,23)(6,24,11)(7,12,21)(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), (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),(5,23,10),(6,24,11),(7,21,12),(8,22,9)], [(1,24),(2,21),(3,22),(4,23),(5,15),(6,16),(7,13),(8,14),(9,19),(10,20),(11,17),(12,18)], [(1,17,16),(2,13,18),(3,19,14),(4,15,20),(5,10,23),(6,24,11),(7,12,21),(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)], [(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(24,204);

On 24 points - transitive group 24T226
Generators in S24
(1 20 21)(2 17 22)(3 18 23)(4 19 24)(5 10 14)(6 11 15)(7 12 16)(8 9 13)
(1 6)(2 7)(3 8)(4 5)(9 23)(10 24)(11 21)(12 22)(13 18)(14 19)(15 20)(16 17)
(1 21 20)(2 17 22)(3 23 18)(4 19 24)(5 14 10)(6 11 15)(7 16 12)(8 9 13)
(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 8)(2 7)(3 6)(4 5)(9 20)(10 19)(11 18)(12 17)(13 21)(14 24)(15 23)(16 22)

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

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

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

G:=TransitiveGroup(24,226);

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

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

S3×C3⋊D4 in GAP, Magma, Sage, TeX

S_3\times C_3\rtimes D_4
% in TeX

G:=Group("S3xC3:D4");
// GroupNames label

G:=SmallGroup(144,153);
// by ID

G=gap.SmallGroup(144,153);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,116,490,3461]);
// Polycyclic

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

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