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G = S3×C3⋊F5  order 360 = 23·32·5

Direct product of S3 and C3⋊F5

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

Aliases: S3×C3⋊F5, D15⋊Dic3, (C3×S3)⋊F5, C5⋊(S3×Dic3), D5.1S32, (S3×D5).S3, C3⋊3(S3×F5), C15⋊3(C4×S3), (C5×S3)⋊Dic3, C15⋊(C2×Dic3), (S3×C15)⋊1C4, C32⋊2(C2×F5), (C3×D15)⋊2C4, (C3×D5).4D6, C32⋊3F5⋊2C2, (C32×D5).3C22, C3⋊1(C2×C3⋊F5), (C3×C3⋊F5)⋊2C2, (C3×C15)⋊5(C2×C4), (C3×S3×D5).1C2, SmallGroup(360,128)

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, C32⋊3F5, 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 12A2B2C3A3B3C4A4B4C4D56A6B6C6D6E1012A12B15A15B15C15D15E30A30B
 size 13515224151545454610102030123030448881212
ρ1111111111111111111111111111    trivial
ρ21-11-111111-1-11-1111-1-11111111-1-1    linear of order 2
ρ31111111-1-1-1-11111111-1-11111111    linear of order 2
ρ41-11-1111-1-1111-1111-1-1-1-111111-1-1    linear of order 2
ρ511-1-1111-iii-i11-1-1-1-11-ii1111111    linear of order 4
ρ61-1-11111-ii-ii1-1-1-1-11-1-ii11111-1-1    linear of order 4
ρ711-1-1111i-i-ii11-1-1-1-11i-i1111111    linear of order 4
ρ81-1-11111i-ii-i1-1-1-1-11-1i-i11111-1-1    linear of order 4
ρ92-22-2-12-10000212-1-11-200-1-12-1-111    orthogonal lifted from D6
ρ1020202-1-1-2-20020-12-1001122-1-1-100    orthogonal lifted from D6
ρ1120202-1-1220020-12-100-1-122-1-1-100    orthogonal lifted from S3
ρ122222-12-100002-12-1-1-1200-1-12-1-1-1-1    orthogonal lifted from S3
ρ132-2-22-12-1000021-211-1-200-1-12-1-111    symplectic lifted from Dic3, Schur index 2
ρ1422-2-2-12-100002-1-2111200-1-12-1-1-1-1    symplectic lifted from Dic3, Schur index 2
ρ1520-202-1-1-2i2i00201-2100i-i22-1-1-100    complex lifted from C4×S3
ρ1620-202-1-12i-2i00201-2100-ii22-1-1-100    complex lifted from C4×S3
ρ174-4004440000-1-40000100-1-1-1-1-111    orthogonal lifted from C2×F5
ρ1844004440000-140000-100-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ194040-2-21000040-2-210000-2-2-21100    orthogonal lifted from S32
ρ2040-40-2-2100004022-10000-2-2-21100    symplectic lifted from S3×Dic3, Schur index 2
ρ214400-24-20000-1-20000-1001-√-15/21+√-15/2-11+√-15/21-√-15/21+√-15/21-√-15/2    complex lifted from C3⋊F5
ρ224-400-24-20000-1200001001-√-15/21+√-15/2-11+√-15/21-√-15/2-1-√-15/2-1+√-15/2    complex lifted from C2×C3⋊F5
ρ234-400-24-20000-1200001001+√-15/21-√-15/2-11-√-15/21+√-15/2-1+√-15/2-1-√-15/2    complex lifted from C2×C3⋊F5
ρ244400-24-20000-1-20000-1001+√-15/21-√-15/2-11-√-15/21+√-15/21-√-15/21+√-15/2    complex lifted from C3⋊F5
ρ2580008-4-40000-200000000-2-211100    orthogonal lifted from S3×F5
ρ268000-4-420000-2000000001-√-151+√-151-1-√-15/2-1+√-15/200    complex faithful
ρ278000-4-420000-2000000001+√-151-√-151-1+√-15/2-1-√-15/200    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)

10000110
01010001
10000100
01011001
01011000
00100010
10100010
00011001
,
00001001
10000100
01010001
00100110
00000010
01001001
00001000
10000010
,
00000110
01010001
10100100
01011001
01011000
00100110
10100000
00011001
,
00100110
00000001
10100110
01000000
01001000
10100100
10000110
01011000
,
00000001
10100000
00001000
00100110
00000110
01011001
01000001
10000100

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
 

Export

Character table of S3×C3⋊F5 in TeX

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