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## G = C2×S3×F5order 240 = 24·3·5

### Direct product of C2, S3 and F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C2×S3×F5
 Chief series C1 — C5 — C15 — C3×D5 — C3×F5 — S3×F5 — C2×S3×F5
 Lower central C15 — C2×S3×F5
 Upper central C1 — C2

Generators and relations for C2×S3×F5
G = < a,b,c,d,e | a2=b3=c2=d5=e4=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d3 >

Subgroups: 488 in 108 conjugacy classes, 42 normal (22 characteristic)
C1, C2, C2, C3, C4, C22, C5, S3, S3, C6, C6, C2×C4, C23, D5, D5, C10, C10, Dic3, C12, D6, D6, C2×C6, C15, C22×C4, F5, F5, D10, D10, C2×C10, C4×S3, C2×Dic3, C2×C12, C22×S3, C5×S3, C3×D5, D15, C30, C2×F5, C2×F5, C22×D5, S3×C2×C4, C3×F5, C3⋊F5, S3×D5, C6×D5, S3×C10, D30, C22×F5, S3×F5, C6×F5, C2×C3⋊F5, C2×S3×D5, C2×S3×F5
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, F5, C4×S3, C22×S3, C2×F5, S3×C2×C4, C22×F5, S3×F5, C2×S3×F5

Character table of C2×S3×F5

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 5 6A 6B 6C 10A 10B 10C 12A 12B 12C 12D 15 30 size 1 1 3 3 5 5 15 15 2 5 5 5 5 15 15 15 15 4 2 10 10 4 12 12 10 10 10 10 8 8 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 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 -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 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 -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 -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 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 -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 1 1 1 linear of order 2 ρ9 1 1 1 1 -1 -1 -1 -1 1 i -i i -i i -i i -i 1 1 -1 -1 1 1 1 i -i i -i 1 1 linear of order 4 ρ10 1 1 -1 -1 -1 -1 1 1 1 -i i -i i i -i i -i 1 1 -1 -1 1 -1 -1 -i i -i i 1 1 linear of order 4 ρ11 1 -1 -1 1 -1 1 1 -1 1 i i -i -i i i -i -i 1 -1 1 -1 -1 1 -1 -i -i i i 1 -1 linear of order 4 ρ12 1 -1 1 -1 -1 1 -1 1 1 -i -i i i i i -i -i 1 -1 1 -1 -1 -1 1 i i -i -i 1 -1 linear of order 4 ρ13 1 -1 1 -1 -1 1 -1 1 1 i i -i -i -i -i i i 1 -1 1 -1 -1 -1 1 -i -i i i 1 -1 linear of order 4 ρ14 1 -1 -1 1 -1 1 1 -1 1 -i -i i i -i -i i i 1 -1 1 -1 -1 1 -1 i i -i -i 1 -1 linear of order 4 ρ15 1 1 -1 -1 -1 -1 1 1 1 i -i i -i -i i -i i 1 1 -1 -1 1 -1 -1 i -i i -i 1 1 linear of order 4 ρ16 1 1 1 1 -1 -1 -1 -1 1 -i i -i i -i i -i i 1 1 -1 -1 1 1 1 -i i -i i 1 1 linear of order 4 ρ17 2 2 0 0 2 2 0 0 -1 -2 -2 -2 -2 0 0 0 0 2 -1 -1 -1 2 0 0 1 1 1 1 -1 -1 orthogonal lifted from D6 ρ18 2 -2 0 0 2 -2 0 0 -1 -2 2 2 -2 0 0 0 0 2 1 1 -1 -2 0 0 -1 1 1 -1 -1 1 orthogonal lifted from D6 ρ19 2 2 0 0 2 2 0 0 -1 2 2 2 2 0 0 0 0 2 -1 -1 -1 2 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ20 2 -2 0 0 2 -2 0 0 -1 2 -2 -2 2 0 0 0 0 2 1 1 -1 -2 0 0 1 -1 -1 1 -1 1 orthogonal lifted from D6 ρ21 2 -2 0 0 -2 2 0 0 -1 -2i -2i 2i 2i 0 0 0 0 2 1 -1 1 -2 0 0 -i -i i i -1 1 complex lifted from C4×S3 ρ22 2 2 0 0 -2 -2 0 0 -1 -2i 2i -2i 2i 0 0 0 0 2 -1 1 1 2 0 0 i -i i -i -1 -1 complex lifted from C4×S3 ρ23 2 -2 0 0 -2 2 0 0 -1 2i 2i -2i -2i 0 0 0 0 2 1 -1 1 -2 0 0 i i -i -i -1 1 complex lifted from C4×S3 ρ24 2 2 0 0 -2 -2 0 0 -1 2i -2i 2i -2i 0 0 0 0 2 -1 1 1 2 0 0 -i i -i i -1 -1 complex lifted from C4×S3 ρ25 4 4 -4 -4 0 0 0 0 4 0 0 0 0 0 0 0 0 -1 4 0 0 -1 1 1 0 0 0 0 -1 -1 orthogonal lifted from C2×F5 ρ26 4 -4 -4 4 0 0 0 0 4 0 0 0 0 0 0 0 0 -1 -4 0 0 1 -1 1 0 0 0 0 -1 1 orthogonal lifted from C2×F5 ρ27 4 -4 4 -4 0 0 0 0 4 0 0 0 0 0 0 0 0 -1 -4 0 0 1 1 -1 0 0 0 0 -1 1 orthogonal lifted from C2×F5 ρ28 4 4 4 4 0 0 0 0 4 0 0 0 0 0 0 0 0 -1 4 0 0 -1 -1 -1 0 0 0 0 -1 -1 orthogonal lifted from F5 ρ29 8 8 0 0 0 0 0 0 -4 0 0 0 0 0 0 0 0 -2 -4 0 0 -2 0 0 0 0 0 0 1 1 orthogonal lifted from S3×F5 ρ30 8 -8 0 0 0 0 0 0 -4 0 0 0 0 0 0 0 0 -2 4 0 0 2 0 0 0 0 0 0 1 -1 orthogonal faithful

Permutation representations of C2×S3×F5
On 30 points - transitive group 30T51
Generators in S30
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)
(1 6 11)(2 7 12)(3 8 13)(4 9 14)(5 10 15)(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 26)(7 27)(8 28)(9 29)(10 30)(11 21)(12 22)(13 23)(14 24)(15 25)
(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 16)(2 18 5 19)(3 20 4 17)(6 21)(7 23 10 24)(8 25 9 22)(11 26)(12 28 15 29)(13 30 14 27)

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

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

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

G:=TransitiveGroup(30,51);

C2×S3×F5 is a maximal subgroup of
D603C4  C3⋊D4⋊F5
C2×S3×F5 is a maximal quotient of
C4⋊F53S3  Dic65F5  (C4×S3)⋊F5  D12.2F5  D12.F5  D60.C4  D15⋊M4(2)  Dic6.F5  C5⋊C8⋊D6  D603C4  C22⋊F5.S3  C5⋊C8.D6  D15⋊C8⋊C2  D152M4(2)  C3⋊D4⋊F5

Matrix representation of C2×S3×F5 in GL6(𝔽61)

 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 60 60 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 60 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 60 60 60 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 11 0 0 0 0 0 0 11 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 60 60 60 60

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,1,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,1,0,0,0,0,60,0,1,0,0,0,60,0,0,1,0,0,60,0,0,0],[11,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,60,0,0,0,0,1,60,0,0,0,0,0,60,0,0,0,1,0,60] >;

C2×S3×F5 in GAP, Magma, Sage, TeX

C_2\times S_3\times F_5
% in TeX

G:=Group("C2xS3xF5");
// GroupNames label

G:=SmallGroup(240,195);
// by ID

G=gap.SmallGroup(240,195);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,490,3461,887]);
// Polycyclic

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

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