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## G = S3×C3⋊F5order 360 = 23·32·5

### Direct product of S3 and C3⋊F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C15 — S3×C3⋊F5
 Chief series C1 — C5 — C15 — C3×C15 — C32×D5 — C3×C3⋊F5 — S3×C3⋊F5
 Lower central C3×C15 — S3×C3⋊F5
 Upper central C1

Generators and relations for S3×C3⋊F5
G = < a,b,c,d,e | a3=b2=c3=d5=e4=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d3 >

Subgroups: 436 in 70 conjugacy classes, 24 normal (all characteristic)
C1, C2, C3, C3, C4, C22, C5, S3, S3, C6, C2×C4, C32, D5, D5, C10, Dic3, C12, D6, C2×C6, C15, C15, C3×S3, C3×S3, C3×C6, F5, D10, C4×S3, C2×Dic3, C5×S3, C3×D5, C3×D5, D15, C30, C3×Dic3, C3⋊Dic3, S3×C6, C2×F5, C3×C15, C3×F5, C3⋊F5, C3⋊F5, S3×D5, C6×D5, S3×Dic3, C32×D5, S3×C15, C3×D15, S3×F5, C2×C3⋊F5, C3×C3⋊F5, C323F5, C3×S3×D5, S3×C3⋊F5
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, F5, C4×S3, C2×Dic3, S32, C2×F5, C3⋊F5, S3×Dic3, S3×F5, C2×C3⋊F5, S3×C3⋊F5

Character table of S3×C3⋊F5

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 5 6A 6B 6C 6D 6E 10 12A 12B 15A 15B 15C 15D 15E 30A 30B size 1 3 5 15 2 2 4 15 15 45 45 4 6 10 10 20 30 12 30 30 4 4 8 8 8 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 -i i i -i 1 1 -1 -1 -1 -1 1 -i i 1 1 1 1 1 1 1 linear of order 4 ρ6 1 -1 -1 1 1 1 1 -i i -i i 1 -1 -1 -1 -1 1 -1 -i i 1 1 1 1 1 -1 -1 linear of order 4 ρ7 1 1 -1 -1 1 1 1 i -i -i i 1 1 -1 -1 -1 -1 1 i -i 1 1 1 1 1 1 1 linear of order 4 ρ8 1 -1 -1 1 1 1 1 i -i i -i 1 -1 -1 -1 -1 1 -1 i -i 1 1 1 1 1 -1 -1 linear of order 4 ρ9 2 -2 2 -2 -1 2 -1 0 0 0 0 2 1 2 -1 -1 1 -2 0 0 -1 -1 2 -1 -1 1 1 orthogonal lifted from D6 ρ10 2 0 2 0 2 -1 -1 -2 -2 0 0 2 0 -1 2 -1 0 0 1 1 2 2 -1 -1 -1 0 0 orthogonal lifted from D6 ρ11 2 0 2 0 2 -1 -1 2 2 0 0 2 0 -1 2 -1 0 0 -1 -1 2 2 -1 -1 -1 0 0 orthogonal lifted from S3 ρ12 2 2 2 2 -1 2 -1 0 0 0 0 2 -1 2 -1 -1 -1 2 0 0 -1 -1 2 -1 -1 -1 -1 orthogonal lifted from S3 ρ13 2 -2 -2 2 -1 2 -1 0 0 0 0 2 1 -2 1 1 -1 -2 0 0 -1 -1 2 -1 -1 1 1 symplectic lifted from Dic3, Schur index 2 ρ14 2 2 -2 -2 -1 2 -1 0 0 0 0 2 -1 -2 1 1 1 2 0 0 -1 -1 2 -1 -1 -1 -1 symplectic lifted from Dic3, Schur index 2 ρ15 2 0 -2 0 2 -1 -1 -2i 2i 0 0 2 0 1 -2 1 0 0 i -i 2 2 -1 -1 -1 0 0 complex lifted from C4×S3 ρ16 2 0 -2 0 2 -1 -1 2i -2i 0 0 2 0 1 -2 1 0 0 -i i 2 2 -1 -1 -1 0 0 complex lifted from C4×S3 ρ17 4 -4 0 0 4 4 4 0 0 0 0 -1 -4 0 0 0 0 1 0 0 -1 -1 -1 -1 -1 1 1 orthogonal lifted from C2×F5 ρ18 4 4 0 0 4 4 4 0 0 0 0 -1 4 0 0 0 0 -1 0 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ19 4 0 4 0 -2 -2 1 0 0 0 0 4 0 -2 -2 1 0 0 0 0 -2 -2 -2 1 1 0 0 orthogonal lifted from S32 ρ20 4 0 -4 0 -2 -2 1 0 0 0 0 4 0 2 2 -1 0 0 0 0 -2 -2 -2 1 1 0 0 symplectic lifted from S3×Dic3, Schur index 2 ρ21 4 4 0 0 -2 4 -2 0 0 0 0 -1 -2 0 0 0 0 -1 0 0 1-√-15/2 1+√-15/2 -1 1+√-15/2 1-√-15/2 1+√-15/2 1-√-15/2 complex lifted from C3⋊F5 ρ22 4 -4 0 0 -2 4 -2 0 0 0 0 -1 2 0 0 0 0 1 0 0 1-√-15/2 1+√-15/2 -1 1+√-15/2 1-√-15/2 -1-√-15/2 -1+√-15/2 complex lifted from C2×C3⋊F5 ρ23 4 -4 0 0 -2 4 -2 0 0 0 0 -1 2 0 0 0 0 1 0 0 1+√-15/2 1-√-15/2 -1 1-√-15/2 1+√-15/2 -1+√-15/2 -1-√-15/2 complex lifted from C2×C3⋊F5 ρ24 4 4 0 0 -2 4 -2 0 0 0 0 -1 -2 0 0 0 0 -1 0 0 1+√-15/2 1-√-15/2 -1 1-√-15/2 1+√-15/2 1-√-15/2 1+√-15/2 complex lifted from C3⋊F5 ρ25 8 0 0 0 8 -4 -4 0 0 0 0 -2 0 0 0 0 0 0 0 0 -2 -2 1 1 1 0 0 orthogonal lifted from S3×F5 ρ26 8 0 0 0 -4 -4 2 0 0 0 0 -2 0 0 0 0 0 0 0 0 1-√-15 1+√-15 1 -1-√-15/2 -1+√-15/2 0 0 complex faithful ρ27 8 0 0 0 -4 -4 2 0 0 0 0 -2 0 0 0 0 0 0 0 0 1+√-15 1-√-15 1 -1+√-15/2 -1-√-15/2 0 0 complex faithful

Permutation representations of S3×C3⋊F5
On 30 points - transitive group 30T83
Generators in S30
(1 14 9)(2 15 10)(3 11 6)(4 12 7)(5 13 8)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)
(1 19)(2 20)(3 16)(4 17)(5 18)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)
(1 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)
(1 19)(2 16 5 17)(3 18 4 20)(6 28 7 30)(8 27 10 26)(9 29)(11 23 12 25)(13 22 15 21)(14 24)

G:=sub<Sym(30)| (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (1,19)(2,20)(3,16)(4,17)(5,18)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30), (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30), (1,19)(2,16,5,17)(3,18,4,20)(6,28,7,30)(8,27,10,26)(9,29)(11,23,12,25)(13,22,15,21)(14,24)>;

G:=Group( (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (1,19)(2,20)(3,16)(4,17)(5,18)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30), (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13)(16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30), (1,19)(2,16,5,17)(3,18,4,20)(6,28,7,30)(8,27,10,26)(9,29)(11,23,12,25)(13,22,15,21)(14,24) );

G=PermutationGroup([[(1,14,9),(2,15,10),(3,11,6),(4,12,7),(5,13,8),(16,21,26),(17,22,27),(18,23,28),(19,24,29),(20,25,30)], [(1,19),(2,20),(3,16),(4,17),(5,18),(6,21),(7,22),(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,29),(15,30)], [(1,9,14),(2,10,15),(3,6,11),(4,7,12),(5,8,13),(16,21,26),(17,22,27),(18,23,28),(19,24,29),(20,25,30)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25),(26,27,28,29,30)], [(1,19),(2,16,5,17),(3,18,4,20),(6,28,7,30),(8,27,10,26),(9,29),(11,23,12,25),(13,22,15,21),(14,24)]])

G:=TransitiveGroup(30,83);

Matrix representation of S3×C3⋊F5 in GL8(𝔽2)

 1 0 0 0 0 1 1 0 0 1 0 1 0 0 0 1 1 0 0 0 0 1 0 0 0 1 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 1 0 0 0 0 1 1 0 0 1
,
 0 0 0 0 1 0 0 1 1 0 0 0 0 1 0 0 0 1 0 1 0 0 0 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0
,
 0 0 0 0 0 1 1 0 0 1 0 1 0 0 0 1 1 0 1 0 0 1 0 0 0 1 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0 0 1 0 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 1 1 0 0 1
,
 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 0 0 0 1 1 0 0 1 0 1 1 0 0 0
,
 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 1 1 0 0 1 0 1 1 0 0 1 0 1 0 0 0 0 0 1 1 0 0 0 0 1 0 0

G:=sub<GL(8,GF(2))| [1,0,1,0,0,0,1,0,0,1,0,1,1,0,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,0,0,1,0,0,0,1,1,0,0,1,1,0,1,0,0,0,0,0,1,0,0,0,0,1,1,0,0,1,0,1,0,0,0,1],[0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,1,0,1,0,0,1,0,0],[0,0,1,0,0,0,1,0,0,1,0,1,1,0,0,0,0,0,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0,0,0,1,1,0,0,1,1,0,1,0,0,1,0,0,1,0,0,0,0,1,0,0,0,1,0,1,0,0,0,1],[0,0,1,0,0,1,1,0,0,0,0,1,1,0,0,1,1,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,1,1,0,1,0,0,1,1,0,1,0,1,0,0,0,1,0,0,1,0,0,0,0,0,0],[0,1,0,0,0,0,0,1,0,0,0,0,0,1,1,0,0,1,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,1,1,0,0,1,0,0,0,1,1,0,0,0,1,0,0,0,0,1,1,0] >;

S3×C3⋊F5 in GAP, Magma, Sage, TeX

S_3\times C_3\rtimes F_5
% in TeX

G:=Group("S3xC3:F5");
// GroupNames label

G:=SmallGroup(360,128);
// by ID

G=gap.SmallGroup(360,128);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-5,24,201,1444,7781,2609]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^2=c^3=d^5=e^4=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,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^3>;
// generators/relations

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