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G = C2×S3×F5  order 240 = 24·3·5

Direct product of C2, S3 and F5

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C2×S3×F5, D30⋊2C4, D10.8D6, C10⋊(C4×S3), C30⋊(C2×C4), (S3×D5)⋊C4, D5⋊(C4×S3), C3⋊F5⋊C22, D15⋊(C2×C4), C15⋊(C22×C4), C6⋊1(C2×F5), (C6×F5)⋊2C2, (S3×C10)⋊2C4, (C3×F5)⋊C22, C3⋊1(C22×F5), (C3×D5).C23, (S3×D5).C22, D5.1(C22×S3), (C6×D5).8C22, C5⋊(S3×C2×C4), (C5×S3)⋊(C2×C4), (C3×D5)⋊(C2×C4), (C2×C3⋊F5)⋊2C2, (C2×S3×D5).2C2, Aut(D30), Hol(C30), SmallGroup(240,195)

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 12A2B2C2D2E2F2G34A4B4C4D4E4F4G4H56A6B6C10A10B10C12A12B12C12D1530
 size 11335515152555515151515421010412121010101088
ρ1111111111111111111111111111111    trivial
ρ211-1-111-1-11-1-1-1-1111111111-1-1-1-1-1-111    linear of order 2
ρ31-11-11-11-11-111-11-1-111-1-11-1-111-1-111-1    linear of order 2
ρ41-1-111-1-1111-1-111-1-111-1-11-11-1-111-11-1    linear of order 2
ρ51-11-11-11-111-1-11-111-11-1-11-1-11-111-11-1    linear of order 2
ρ61-1-111-1-111-111-1-111-11-1-11-11-11-1-111-1    linear of order 2
ρ7111111111-1-1-1-1-1-1-1-11111111-1-1-1-111    linear of order 2
ρ811-1-111-1-111111-1-1-1-111111-1-1111111    linear of order 2
ρ91111-1-1-1-11i-ii-ii-ii-i11-1-1111i-ii-i11    linear of order 4
ρ1011-1-1-1-1111-ii-iii-ii-i11-1-11-1-1-ii-ii11    linear of order 4
ρ111-1-11-111-11ii-i-iii-i-i1-11-1-11-1-i-iii1-1    linear of order 4
ρ121-11-1-11-111-i-iiiii-i-i1-11-1-1-11ii-i-i1-1    linear of order 4
ρ131-11-1-11-111ii-i-i-i-iii1-11-1-1-11-i-iii1-1    linear of order 4
ρ141-1-11-111-11-i-iii-i-iii1-11-1-11-1ii-i-i1-1    linear of order 4
ρ1511-1-1-1-1111i-ii-i-ii-ii11-1-11-1-1i-ii-i11    linear of order 4
ρ161111-1-1-1-11-ii-ii-ii-ii11-1-1111-ii-ii11    linear of order 4
ρ1722002200-1-2-2-2-200002-1-1-12001111-1-1    orthogonal lifted from D6
ρ182-2002-200-1-222-20000211-1-200-111-1-11    orthogonal lifted from D6
ρ1922002200-1222200002-1-1-1200-1-1-1-1-1-1    orthogonal lifted from S3
ρ202-2002-200-12-2-220000211-1-2001-1-11-11    orthogonal lifted from D6
ρ212-200-2200-1-2i-2i2i2i000021-11-200-i-iii-11    complex lifted from C4×S3
ρ222200-2-200-1-2i2i-2i2i00002-111200i-ii-i-1-1    complex lifted from C4×S3
ρ232-200-2200-12i2i-2i-2i000021-11-200ii-i-i-11    complex lifted from C4×S3
ρ242200-2-200-12i-2i2i-2i00002-111200-ii-ii-1-1    complex lifted from C4×S3
ρ2544-4-40000400000000-1400-1110000-1-1    orthogonal lifted from C2×F5
ρ264-4-440000400000000-1-4001-110000-11    orthogonal lifted from C2×F5
ρ274-44-40000400000000-1-40011-10000-11    orthogonal lifted from C2×F5
ρ2844440000400000000-1400-1-1-10000-1-1    orthogonal lifted from F5
ρ2988000000-400000000-2-400-200000011    orthogonal lifted from S3×F5
ρ308-8000000-400000000-240020000001-1    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
 D60⋊3C4  C3⋊D4⋊F5
C2×S3×F5 is a maximal quotient of
 C4⋊F5⋊3S3  Dic6⋊5F5  (C4×S3)⋊F5  D12.2F5  D12.F5  D60.C4  D15⋊M4(2)  Dic6.F5  C5⋊C8⋊D6  D60⋊3C4  C22⋊F5.S3  C5⋊C8.D6  D15⋊C8⋊C2  D15⋊2M4(2)  C3⋊D4⋊F5

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

6000000
0600000
001000
000100
000010
000001
,
60600000
100000
001000
000100
000010
000001
,
6000000
110000
001000
000100
000010
000001
,
100000
010000
0060606060
001000
000100
000010
,
1100000
0110000
001000
000001
000100
0060606060

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
 

Export

Character table of C2×S3×F5 in TeX

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