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G = C5⋊S3≀C2  order 360 = 23·32·5

The semidirect product of C5 and S3≀C2 acting via S3≀C2/S32=C2

non-abelian, soluble, monomial

Aliases: C5⋊2S3≀C2, (S32)⋊D5, (C3×C15)⋊1D4, C32⋊(C5⋊D4), D15⋊S3⋊1C2, C3⋊S3.1D10, C32⋊Dic5⋊2C2, (C5×S32)⋊3C2, (C5×C3⋊S3).2C22, SmallGroup(360,133)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C5×C3⋊S3 — C5⋊S3≀C2
Chief series
C1 — C5 — C3×C15 — C5×C3⋊S3 — D15⋊S3 — C5⋊S3≀C2
Lower central
C3×C15 — C5×C3⋊S3 — C5⋊S3≀C2
Upper central
C1

Generators and relations for C5⋊S3≀C2
 G = < a,b,c,d,e | a5=b3=c3=d4=e2=1, ab=ba, ac=ca, dad-1=eae=a-1, bc=cb, dbd-1=c, ebe=dcd-1=b-1, ce=ec, ede=d-1 >

C1
6C2
9C2
30C2
2C3
2C3
C5
9C22
45C22
45C4
2S3
6S3
6C6
6S3
10S3
30C6
C32
6D5
6C10
9C10
2C15
2C15
45D4
6D6
30D6
C3⋊S3
2C3×S3
10C3×S3
9D10
9Dic5
9C2×C10
2C5×S3
2D15
6C3×D5
6C30
6C5×S3
6C5×S3
C3×C15
S32
5S32
5C32⋊C4
9C5⋊D4
6S3×C10
6S3×D5
C5×C3⋊S3
2C3×D15
2S3×C15
5S3≀C2
D15⋊S3
C32⋊Dic5
C5×S32
C5⋊S3≀C2

Character table of C5⋊S3≀C2

 class 12A2B2C3A3B45A5B6A6B10A10B10C10D10E10F15A15B15C15D15E15F30A30B30C30D
 size 1693044902212606666181844448812121212
ρ1111111111111111111111111111    trivial
ρ21-11111-111-11-1-1-1-111111111-1-1-1-1    linear of order 2
ρ31-11-111111-1-1-1-1-1-111111111-1-1-1-1    linear of order 2
ρ4111-111-1111-11111111111111111    linear of order 2
ρ520-2022022000000-2-22222220000    orthogonal lifted from D4
ρ62-220220-1-√5/2-1+√5/2-201-√5/21-√5/21+√5/21+√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/21+√5/21+√5/21-√5/21-√5/2    orthogonal lifted from D10
ρ72220220-1+√5/2-1-√5/220-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2    orthogonal lifted from D5
ρ82220220-1-√5/2-1+√5/220-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2    orthogonal lifted from D5
ρ92-220220-1+√5/2-1-√5/2-201+√5/21+√5/21-√5/21-√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/21-√5/21-√5/21+√5/21+√5/2    orthogonal lifted from D10
ρ1020-20220-1+√5/2-1-√5/200ζ53-ζ52-ζ53+ζ52ζ54-ζ5-ζ54+ζ51-√5/21+√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/2ζ54-ζ5-ζ54+ζ5ζ53-ζ52-ζ53+ζ52    complex lifted from C5⋊D4
ρ1120-20220-1+√5/2-1-√5/200-ζ53+ζ52ζ53-ζ52-ζ54+ζ5ζ54-ζ51-√5/21+√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/2-ζ54+ζ5ζ54-ζ5-ζ53+ζ52ζ53-ζ52    complex lifted from C5⋊D4
ρ1220-20220-1-√5/2-1+√5/200-ζ54+ζ5ζ54-ζ5ζ53-ζ52-ζ53+ζ521+√5/21-√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/2ζ53-ζ52-ζ53+ζ52-ζ54+ζ5ζ54-ζ5    complex lifted from C5⋊D4
ρ1320-20220-1-√5/2-1+√5/200ζ54-ζ5-ζ54+ζ5-ζ53+ζ52ζ53-ζ521+√5/21-√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/2-1+√5/2-1-√5/2-ζ53+ζ52ζ53-ζ52ζ54-ζ5-ζ54+ζ5    complex lifted from C5⋊D4
ρ14400-21-204401000000-2-2-2-2110000    orthogonal lifted from S3≀C2
ρ154-200-2104410-2-2-2-2001111-2-21111    orthogonal lifted from S3≀C2
ρ1640021-20440-1000000-2-2-2-2110000    orthogonal lifted from S3≀C2
ρ174200-21044-102222001111-2-2-1-1-1-1    orthogonal lifted from S3≀C2
ρ184-200-210-1+√5-1-√510-2ζ53-2ζ52-2ζ54-2ζ5002ζ53-ζ522ζ54-ζ5-ζ53+2ζ52-ζ54+2ζ51+√5/21-√5/2ζ54ζ5ζ53ζ52    complex faithful
ρ194200-210-1-√5-1+√5-102ζ542ζ52ζ522ζ53002ζ54-ζ5-ζ53+2ζ52-ζ54+2ζ52ζ53-ζ521-√5/21+√5/2-ζ52-ζ53-ζ54-ζ5    complex faithful
ρ204200-210-1-√5-1+√5-102ζ52ζ542ζ532ζ5200-ζ54+2ζ52ζ53-ζ522ζ54-ζ5-ζ53+2ζ521-√5/21+√5/2-ζ53-ζ52-ζ5-ζ54    complex faithful
ρ214200-210-1+√5-1-√5-102ζ532ζ522ζ542ζ5002ζ53-ζ522ζ54-ζ5-ζ53+2ζ52-ζ54+2ζ51+√5/21-√5/2-ζ54-ζ5-ζ53-ζ52    complex faithful
ρ224-200-210-1-√5-1+√510-2ζ54-2ζ5-2ζ52-2ζ53002ζ54-ζ5-ζ53+2ζ52-ζ54+2ζ52ζ53-ζ521-√5/21+√5/2ζ52ζ53ζ54ζ5    complex faithful
ρ234-200-210-1-√5-1+√510-2ζ5-2ζ54-2ζ53-2ζ5200-ζ54+2ζ52ζ53-ζ522ζ54-ζ5-ζ53+2ζ521-√5/21+√5/2ζ53ζ52ζ5ζ54    complex faithful
ρ244-200-210-1+√5-1-√510-2ζ52-2ζ53-2ζ5-2ζ5400-ζ53+2ζ52-ζ54+2ζ52ζ53-ζ522ζ54-ζ51+√5/21-√5/2ζ5ζ54ζ52ζ53    complex faithful
ρ254200-210-1+√5-1-√5-102ζ522ζ532ζ52ζ5400-ζ53+2ζ52-ζ54+2ζ52ζ53-ζ522ζ54-ζ51+√5/21-√5/2-ζ5-ζ54-ζ52-ζ53    complex faithful
ρ2680002-40-2+2√5-2-2√5000000001+√51-√51+√51-√5-1-√5/2-1+√5/20000    orthogonal faithful
ρ2780002-40-2-2√5-2+2√5000000001-√51+√51-√51+√5-1+√5/2-1-√5/20000    orthogonal faithful

Permutation representations of C5⋊S3≀C2
►On 30 points - transitive group 30T96
Generators in S30
(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)

(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30)

(1 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)

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

(2 5)(3 4)(6 7)(8 10)(11 12)(13 15)(16 17)(18 20)(21 27)(22 26)(23 30)(24 29)(25 28)
 
G:=sub<Sym(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), (16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13), (1,19)(2,18)(3,17)(4,16)(5,20)(6,22,11,27)(7,21,12,26)(8,25,13,30)(9,24,14,29)(10,23,15,28), (2,5)(3,4)(6,7)(8,10)(11,12)(13,15)(16,17)(18,20)(21,27)(22,26)(23,30)(24,29)(25,28)>;
 
G:=Group( (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), (16,21,26)(17,22,27)(18,23,28)(19,24,29)(20,25,30), (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13), (1,19)(2,18)(3,17)(4,16)(5,20)(6,22,11,27)(7,21,12,26)(8,25,13,30)(9,24,14,29)(10,23,15,28), (2,5)(3,4)(6,7)(8,10)(11,12)(13,15)(16,17)(18,20)(21,27)(22,26)(23,30)(24,29)(25,28) );
 
G=PermutationGroup([(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)], [(16,21,26),(17,22,27),(18,23,28),(19,24,29),(20,25,30)], [(1,9,14),(2,10,15),(3,6,11),(4,7,12),(5,8,13)], [(1,19),(2,18),(3,17),(4,16),(5,20),(6,22,11,27),(7,21,12,26),(8,25,13,30),(9,24,14,29),(10,23,15,28)], [(2,5),(3,4),(6,7),(8,10),(11,12),(13,15),(16,17),(18,20),(21,27),(22,26),(23,30),(24,29),(25,28)])
 
G:=TransitiveGroup(30,96);
 

►On 30 points - transitive group 30T100
Generators in S30
(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 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 14 9)(2 15 10)(3 11 6)(4 12 7)(5 13 8)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)

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

(1 16)(2 20)(3 19)(4 18)(5 17)(6 24)(7 23)(8 22)(9 21)(10 25)(11 29)(12 28)(13 27)(14 26)(15 30)
 
G:=sub<Sym(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), (1,14,9)(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), (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25), (1,16)(2,20)(3,19)(4,18)(5,17)(6,29,11,24)(7,28,12,23)(8,27,13,22)(9,26,14,21)(10,30,15,25), (1,16)(2,20)(3,19)(4,18)(5,17)(6,24)(7,23)(8,22)(9,21)(10,25)(11,29)(12,28)(13,27)(14,26)(15,30)>;
 
G:=Group( (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), (1,14,9)(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), (1,14,9)(2,15,10)(3,11,6)(4,12,7)(5,13,8)(16,26,21)(17,27,22)(18,28,23)(19,29,24)(20,30,25), (1,16)(2,20)(3,19)(4,18)(5,17)(6,29,11,24)(7,28,12,23)(8,27,13,22)(9,26,14,21)(10,30,15,25), (1,16)(2,20)(3,19)(4,18)(5,17)(6,24)(7,23)(8,22)(9,21)(10,25)(11,29)(12,28)(13,27)(14,26)(15,30) );
 
G=PermutationGroup([(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)], [(1,14,9),(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)], [(1,14,9),(2,15,10),(3,11,6),(4,12,7),(5,13,8),(16,26,21),(17,27,22),(18,28,23),(19,29,24),(20,30,25)], [(1,16),(2,20),(3,19),(4,18),(5,17),(6,29,11,24),(7,28,12,23),(8,27,13,22),(9,26,14,21),(10,30,15,25)], [(1,16),(2,20),(3,19),(4,18),(5,17),(6,24),(7,23),(8,22),(9,21),(10,25),(11,29),(12,28),(13,27),(14,26),(15,30)])
 
G:=TransitiveGroup(30,100);
 

Matrix representation of C5⋊S3≀C2 ►in GL6(𝔽61)

3400000
090000
001000
000100
000010
000001
,
100000
010000
001000
000100
00005915
0000121
,
100000
010000
00591500
0012100
000010
000001
,
0600000
100000
0000600
0000121
0060000
0006000
,
0600000
6000000
0060000
0006000
0000600
0000121

G:=sub<GL(6,GF(61))| [34,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,59,12,0,0,0,0,15,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,59,12,0,0,0,0,15,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,60,12,0,0,0,0,0,1,0,0],[0,60,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,12,0,0,0,0,0,1] >;
 

C5⋊S3≀C2 in GAP, Magma, Sage, TeX 

C_5\rtimes S_3\wr C_2
 
% in TeX
 
G:=Group("C5:S3wrC2");
 
// GroupNames label
 
G:=SmallGroup(360,133);
 
// by ID
 
G=gap.SmallGroup(360,133);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-3,3,-5,73,579,201,111,244,376,10373]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^5=b^3=c^3=d^4=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e=a^-1,b*c=c*b,d*b*d^-1=c,e*b*e=d*c*d^-1=b^-1,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

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