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G = C32⋊F5⋊C2  order 360 = 23·32·5

The semidirect product of C32⋊F5 and C2 acting faithfully

metabelian, soluble, monomial, A-group

Aliases: C32⋊F5⋊C2, C3⋊S3⋊2F5, D5⋊(C32⋊C4), C32⋊1(C2×F5), C3⋊D15.C22, (C32×D5)⋊2C4, C5⋊1(C2×C32⋊C4), (C5×C3⋊S3)⋊1C4, (C3×C15)⋊2(C2×C4), (D5×C3⋊S3).2C2, SmallGroup(360,131)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C15 — C32⋊F5⋊C2
Chief series
C1 — C5 — C3×C15 — C3⋊D15 — C32⋊F5 — C32⋊F5⋊C2
Lower central
C3×C15 — C32⋊F5⋊C2
Upper central
C1

Generators and relations for C32⋊F5⋊C2
 G = < a,b,c,d,e | a3=b3=c5=d4=e2=1, dbd-1=ab=ba, ac=ca, dad-1=a-1b, eae=a-1, bc=cb, ebe=b-1, dcd-1=c3, ce=ec, de=ed >

C1
5C2
9C2
45C2
2C3
2C3
C5
45C4
45C22
45C4
6S3
6S3
10C6
10C6
30S3
30S3
C32
D5
9D5
9C10
2C15
2C15
45C2×C4
30D6
30D6
C3⋊S3
5C3×C6
5C3⋊S3
9D10
9F5
9F5
2C3×D5
2C3×D5
6C5×S3
6D15
6C5×S3
6D15
C3×C15
5C32⋊C4
5C2×C3⋊S3
5C32⋊C4
9C2×F5
6S3×D5
6S3×D5
C3⋊D15
C5×C3⋊S3
C32×D5
5C2×C32⋊C4
C32⋊F5
C32⋊F5
D5×C3⋊S3
C32⋊F5⋊C2

Character table of C32⋊F5⋊C2

 class 12A2B2C3A3B4A4B4C4D56A6B1015A15B15C15D
 size 15945444545454542020368888
ρ1111111111111111111    trivial
ρ21-1-1111-111-11-1-1-11111    linear of order 2
ρ3111111-1-1-1-111111111    linear of order 2
ρ41-1-11111-1-111-1-1-11111    linear of order 2
ρ51-11-111i-ii-i1-1-111111    linear of order 4
ρ611-1-111-i-iii111-11111    linear of order 4
ρ711-1-111ii-i-i111-11111    linear of order 4
ρ81-11-111-ii-ii1-1-111111    linear of order 4
ρ94-4001-2000042-10-21-21    orthogonal lifted from C2×C32⋊C4
ρ104-400-2100004-1201-21-2    orthogonal lifted from C2×C32⋊C4
ρ114040440000-100-1-1-1-1-1    orthogonal lifted from F5
ρ124400-21000041-201-21-2    orthogonal lifted from C32⋊C4
ρ1344001-200004-210-21-21    orthogonal lifted from C32⋊C4
ρ1440-40440000-1001-1-1-1-1    orthogonal lifted from C2×F5
ρ1580002-40000-20001-1-3√5/21-1+3√5/2    orthogonal faithful
ρ1680002-40000-20001-1+3√5/21-1-3√5/2    orthogonal faithful
ρ178000-420000-2000-1+3√5/21-1-3√5/21    orthogonal faithful
ρ188000-420000-2000-1-3√5/21-1+3√5/21    orthogonal faithful

Permutation representations of C32⋊F5⋊C2
►On 30 points - transitive group 30T97
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 9 14)(2 10 15)(3 6 11)(4 7 12)(5 8 13)

(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 27 12 24)(7 29 11 22)(8 26 15 25)(9 28 14 23)(10 30 13 21)

(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), (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13), (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)(6,27,12,24)(7,29,11,22)(8,26,15,25)(9,28,14,23)(10,30,13,21), (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), (1,9,14)(2,10,15)(3,6,11)(4,7,12)(5,8,13), (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)(6,27,12,24)(7,29,11,22)(8,26,15,25)(9,28,14,23)(10,30,13,21), (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)], [(1,9,14),(2,10,15),(3,6,11),(4,7,12),(5,8,13)], [(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),(6,27,12,24),(7,29,11,22),(8,26,15,25),(9,28,14,23),(10,30,13,21)], [(6,11),(7,12),(8,13),(9,14),(10,15),(21,26),(22,27),(23,28),(24,29),(25,30)]])
 
G:=TransitiveGroup(30,97);
 

Matrix representation of C32⋊F5⋊C2 ►in GL8(𝔽61)

10000000
01000000
00100000
00010000
00001000
00000100
00000015
0000113659
,
10000000
01000000
00100000
00010000
000006000
000016000
000005615
0000113659
,
601000000
600100000
600010000
600000000
00001000
00000100
00000010
00000001
,
6016000000
5910600000
6001590000
0601600000
00000001
0000113660
000000600
00000100
,
600000000
060000000
006000000
000600000
000006000
000060000
000000600
00006060251

G:=sub<GL(8,GF(61))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,0,1,36,0,0,0,0,0,0,5,59],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,60,60,56,1,0,0,0,0,0,0,1,36,0,0,0,0,0,0,5,59],[60,60,60,60,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[60,59,60,0,0,0,0,0,1,1,0,60,0,0,0,0,60,0,1,1,0,0,0,0,0,60,59,60,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,36,60,0,0,0,0,0,1,60,0,0],[60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,60,0,0,0,0,60,0,0,60,0,0,0,0,0,0,60,25,0,0,0,0,0,0,0,1] >;
 

C32⋊F5⋊C2 in GAP, Magma, Sage, TeX 

C_3^2\rtimes F_5\rtimes C_2
 
% in TeX
 
G:=Group("C3^2:F5:C2");
 
// GroupNames label
 
G:=SmallGroup(360,131);
 
// by ID
 
G=gap.SmallGroup(360,131);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-3,3,-5,24,963,201,111,964,730,376,7781,2609]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^5=d^4=e^2=1,d*b*d^-1=a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b,e*a*e=a^-1,b*c=c*b,e*b*e=b^-1,d*c*d^-1=c^3,c*e=e*c,d*e=e*d>;
 
// generators/relations
 

Export

Subgroup lattice of C32⋊F5⋊C2 in TeX
Character table of C32⋊F5⋊C2 in TeX

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