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G = C3⋊F5⋊S3  order 360 = 23·32·5

The semidirect product of C3⋊F5 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial, A-group

Aliases: C3⋊F5⋊S3, D5.2S32, C3⋊S3⋊1F5, C3⋊2(S3×F5), C15⋊2(C4×S3), C3⋊D15⋊3C4, C5⋊(C6.D6), C32⋊3(C2×F5), (C3×D5).3D6, (C32×D5).4C22, (C3×C3⋊F5)⋊3C2, (C5×C3⋊S3)⋊3C4, (C3×C15)⋊6(C2×C4), (D5×C3⋊S3).4C2, SmallGroup(360,129)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for C3⋊F5⋊S3
 G = < a,b,c,d,e | a3=b5=c4=d3=e2=1, ab=ba, cac-1=eae=a-1, ad=da, cbc-1=b3, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 548 in 74 conjugacy classes, 21 normal (11 characteristic)
C1, C2, C3, C3, C4, C22, C5, S3, C6, C2×C4, C32, D5, D5, C10, Dic3, C12, D6, C15, C15, C3⋊S3, C3⋊S3, C3×C6, F5, D10, C4×S3, C5×S3, C3×D5, C3×D5, D15, C3×Dic3, C2×C3⋊S3, C2×F5, C3×C15, C3×F5, C3⋊F5, S3×D5, C6.D6, C32×D5, C5×C3⋊S3, C3⋊D15, S3×F5, C3×C3⋊F5, D5×C3⋊S3, C3⋊F5⋊S3
Quotients: C1, C2, C4, C22, S3, C2×C4, D6, F5, C4×S3, S32, C2×F5, C6.D6, S3×F5, C3⋊F5⋊S3

Character table of C3⋊F5⋊S3

 class 12A2B2C3A3B3C4A4B4C4D56A6B6C1012A12B12C12D15A15B15C15D
 size 1594522415151515410102036303030308888
ρ1111111111111111111111111    trivial
ρ211-1-1111-111-11111-1-11-111111    linear of order 2
ρ31111111-1-1-1-111111-1-1-1-11111    linear of order 2
ρ411-1-11111-1-111111-11-11-11111    linear of order 2
ρ51-1-11111ii-i-i1-1-1-1-1i-i-ii1111    linear of order 4
ρ61-11-1111-ii-ii1-1-1-11-i-iii1111    linear of order 4
ρ71-1-11111-i-iii1-1-1-1-1-iii-i1111    linear of order 4
ρ81-11-1111i-ii-i1-1-1-11ii-i-i1111    linear of order 4
ρ922002-1-1-200-222-1-101010-1-1-12    orthogonal lifted from D6
ρ102200-12-10-2-202-12-1001012-1-1-1    orthogonal lifted from D6
ρ112200-12-102202-12-100-10-12-1-1-1    orthogonal lifted from S3
ρ1222002-1-1200222-1-10-10-10-1-1-12    orthogonal lifted from S3
ρ132-200-12-102i-2i021-2100i0-i2-1-1-1    complex lifted from C4×S3
ρ142-200-12-10-2i2i021-2100-i0i2-1-1-1    complex lifted from C4×S3
ρ152-2002-1-1-2i002i2-2110i0-i0-1-1-12    complex lifted from C4×S3
ρ162-2002-1-12i00-2i2-2110-i0i0-1-1-12    complex lifted from C4×S3
ρ1740404440000-1000-10000-1-1-1-1    orthogonal lifted from F5
ρ1840-404440000-100010000-1-1-1-1    orthogonal lifted from C2×F5
ρ194400-2-2100004-2-2100000-211-2    orthogonal lifted from S32
ρ204-400-2-210000422-100000-211-2    orthogonal lifted from C6.D6
ρ218000-48-40000-200000000-2111    orthogonal lifted from S3×F5
ρ2280008-4-40000-200000000111-2    orthogonal lifted from S3×F5
ρ238000-4-420000-2000000001-1+3√5/2-1-3√5/21    orthogonal faithful
ρ248000-4-420000-2000000001-1-3√5/2-1+3√5/21    orthogonal faithful

Permutation representations of C3⋊F5⋊S3
►On 30 points - transitive group 30T86
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 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 18)(2 20 5 16)(3 17 4 19)(6 22 7 24)(8 21 10 25)(9 23)(11 27 12 29)(13 26 15 30)(14 28)

(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)

(6 11)(7 12)(8 13)(9 14)(10 15)(21 26)(22 27)(23 28)(24 29)(25 30)
 
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), (1,2,3,4,5)(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), (1,18)(2,20,5,16)(3,17,4,19)(6,22,7,24)(8,21,10,25)(9,23)(11,27,12,29)(13,26,15,30)(14,28), (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), (6,11)(7,12)(8,13)(9,14)(10,15)(21,26)(22,27)(23,28)(24,29)(25,30)>;
 
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), (1,2,3,4,5)(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), (1,18)(2,20,5,16)(3,17,4,19)(6,22,7,24)(8,21,10,25)(9,23)(11,27,12,29)(13,26,15,30)(14,28), (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), (6,11)(7,12)(8,13)(9,14)(10,15)(21,26)(22,27)(23,28)(24,29)(25,30) );
 
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)], [(1,2,3,4,5),(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)], [(1,18),(2,20,5,16),(3,17,4,19),(6,22,7,24),(8,21,10,25),(9,23),(11,27,12,29),(13,26,15,30),(14,28)], [(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)], [(6,11),(7,12),(8,13),(9,14),(10,15),(21,26),(22,27),(23,28),(24,29),(25,30)]])
 
G:=TransitiveGroup(30,86);
 

Matrix representation of C3⋊F5⋊S3 ►in GL8(𝔽61)

10000000
01000000
006010000
006000000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
000060100
000060010
000060001
000060000
,
110000000
011000000
000500000
005000000
00001051540
0000351010
0000100513
00000545110
,
01000000
6060000000
00100000
00010000
00001000
00000100
00000010
00000001
,
060000000
600000000
000600000
006000000
000060000
000006000
000000600
000000060

G:=sub<GL(8,GF(61))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,60,60,60,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,50,0,0,0,0,0,0,50,0,0,0,0,0,0,0,0,0,10,3,10,0,0,0,0,0,51,51,0,54,0,0,0,0,54,0,51,51,0,0,0,0,0,10,3,10],[0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60] >;
 

C3⋊F5⋊S3 in GAP, Magma, Sage, TeX 

C_3\rtimes F_5\rtimes S_3
 
% in TeX
 
G:=Group("C3:F5:S3");
 
// GroupNames label
 
G:=SmallGroup(360,129);
 
// by ID
 
G=gap.SmallGroup(360,129);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-3,-3,-5,24,387,201,730,7781,2609]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^5=c^4=d^3=e^2=1,a*b=b*a,c*a*c^-1=e*a*e=a^-1,a*d=d*a,c*b*c^-1=b^3,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

Export

Character table of C3⋊F5⋊S3 in TeX

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