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## G = S3×C23order 48 = 24·3

### Direct product of C23 and S3

Aliases: S3×C23, C3⋊C24, C6⋊C23, (C22×C6)⋊3C2, (C2×C6)⋊4C22, SmallGroup(48,51)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C23
 Chief series C1 — C3 — S3 — D6 — C22×S3 — S3×C23
 Lower central C3 — S3×C23
 Upper central C1 — C23

Generators and relations for S3×C23
G = < a,b,c,d,e | a2=b2=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 236 in 134 conjugacy classes, 83 normal (5 characteristic)
C1, C2 [×7], C2 [×8], C3, C22 [×7], C22 [×28], S3 [×8], C6 [×7], C23, C23 [×14], D6 [×28], C2×C6 [×7], C24, C22×S3 [×14], C22×C6, S3×C23
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C24, C22×S3 [×7], S3×C23

Character table of S3×C23

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 3 6A 6B 6C 6D 6E 6F 6G size 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 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 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 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 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 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 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 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 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 linear of order 2 ρ9 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 ρ10 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 ρ11 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 ρ12 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 ρ13 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 ρ14 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 ρ15 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 ρ16 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 ρ17 2 -2 2 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 -1 1 1 -1 1 -1 1 -1 orthogonal lifted from D6 ρ18 2 2 -2 2 -2 2 -2 -2 0 0 0 0 0 0 0 0 -1 -1 -1 1 1 -1 1 1 orthogonal lifted from D6 ρ19 2 -2 2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 -1 1 -1 1 1 1 -1 -1 orthogonal lifted from D6 ρ20 2 2 -2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 -1 -1 1 -1 1 1 -1 1 orthogonal lifted from D6 ρ21 2 2 2 -2 2 -2 -2 -2 0 0 0 0 0 0 0 0 -1 -1 1 1 -1 1 1 -1 orthogonal lifted from D6 ρ22 2 -2 -2 -2 2 2 2 -2 0 0 0 0 0 0 0 0 -1 1 -1 -1 -1 1 1 1 orthogonal lifted from D6 ρ23 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ24 2 -2 -2 2 2 -2 -2 2 0 0 0 0 0 0 0 0 -1 1 1 1 -1 -1 -1 1 orthogonal lifted from D6

Permutation representations of S3×C23
On 24 points - transitive group 24T30
Generators in S24
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)
(1 10)(2 11)(3 12)(4 7)(5 8)(6 9)(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)
(1 4)(2 5)(3 6)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(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 16)(2 18)(3 17)(4 13)(5 15)(6 14)(7 22)(8 24)(9 23)(10 19)(11 21)(12 20)

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

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

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

G:=TransitiveGroup(24,30);

S3×C23 is a maximal subgroup of   D6⋊D4  C232D6
S3×C23 is a maximal quotient of   D46D6  Q8.15D6  D4○D12  Q8○D12

Matrix representation of S3×C23 in GL4(ℤ) generated by

 1 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 -1
,
 -1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 -1 -1 0 0 1 0
,
 -1 0 0 0 0 1 0 0 0 0 1 0 0 0 -1 -1
G:=sub<GL(4,Integers())| [1,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,-1],[-1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,-1,1,0,0,-1,0],[-1,0,0,0,0,1,0,0,0,0,1,-1,0,0,0,-1] >;

S3×C23 in GAP, Magma, Sage, TeX

S_3\times C_2^3
% in TeX

G:=Group("S3xC2^3");
// GroupNames label

G:=SmallGroup(48,51);
// by ID

G=gap.SmallGroup(48,51);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-3,804]);
// Polycyclic

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

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