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G = C62:5D6  order 432 = 24·33

5th semidirect product of C62 and D6 acting faithfully

non-abelian, soluble, monomial, rational

Aliases: C62:5D6, C3:S3:S4, C3:S4:S3, (C3xA4):D6, C32:(C2xS4), C3.2(S3xS4), C62:S3:C2, C62:C6:C2, C32:S4:C2, C32:A4:C22, C22:(C32:D6), (C2xC6).2S32, (C22xC3:S3):4S3, SmallGroup(432,523)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2xC6 — C32:A4 — C62:5D6
Chief series
C1 — C22 — C2xC6 — C62 — C32:A4 — C62:C6 — C62:5D6
Lower central
C32:A4 — C62:5D6
Upper central
C1

Generators and relations for C62:5D6
 G = < a,b,c,d | a6=b6=c6=d2=1, ab=ba, cac-1=a2b-1, dad=a-1b-1, cbc-1=a3b-1, bd=db, dcd=c-1 >

Subgroups: 1289 in 134 conjugacy classes, 15 normal (all characteristic)
C1, C2, C3, C3, C4, C22, C22, S3, C6, C2xC4, D4, C23, C32, C32, Dic3, C12, A4, D6, C2xC6, C2xC6, C2xD4, C3xS3, C3:S3, C3:S3, C3xC6, C4xS3, D12, C3:D4, C3xD4, S4, C2xA4, C22xS3, He3, C3xDic3, S32, C3xA4, C3xA4, S3xC6, C2xC3:S3, C62, S3xD4, C2xS4, C32:C6, He3:C2, C6.D6, C3:D12, C3xC3:D4, C3xS4, C3:S4, S3xA4, C2xS32, C22xC3:S3, C32:D6, C32:A4, Dic3:D6, S3xS4, C62:S3, C32:S4, C62:C6, C62:5D6
Quotients: C1, C2, C22, S3, D6, S4, S32, C2xS4, C32:D6, S3xS4, C62:5D6

Character table of C62:5D6

 class 12A2B2C2D2E3A3B3C3D4A4B6A6B6C6D6E6F12A12B
 size 139181827262448181866123636723636
ρ111111111111111111111    trivial
ρ211-11-1-11111-111111-1-1-11    linear of order 2
ρ3111-1-111111-1-1111-1-11-1-1    linear of order 2
ρ411-1-11-111111-1111-11-11-1    linear of order 2
ρ522-200-222-1-10022200100    orthogonal lifted from D6
ρ622200222-1-10022200-100    orthogonal lifted from S3
ρ72200-202-12-1-202-1-101010    orthogonal lifted from D6
ρ82200202-12-1202-1-10-10-10    orthogonal lifted from S3
ρ93-13-1-1-1330011-1-1-1-1-1011    orthogonal lifted from S4
ρ103-1-31-1133001-1-1-1-11-101-1    orthogonal lifted from C2xS4
ρ113-1311-13300-1-1-1-1-1110-1-1    orthogonal lifted from S4
ρ123-1-3-1113300-11-1-1-1-110-11    orthogonal lifted from C2xS4
ρ134400004-2-21004-2-200000    orthogonal lifted from S32
ρ146-20200-30000-214-2-10001    orthogonal faithful
ρ156-200206-300-20-2110-1010    orthogonal lifted from S3xS4
ρ166-200-206-30020-211010-10    orthogonal lifted from S3xS4
ρ17660200-300002-300-1000-1    orthogonal lifted from C32:D6
ρ186-20-200-30000214-21000-1    orthogonal faithful
ρ19660-200-30000-2-30010001    orthogonal lifted from C32:D6
ρ2012-40000-6000002-4200000    orthogonal faithful

Permutation representations of C62:5D6
►On 18 points - transitive group 18T152
Generators in S18
(1 2)(3 4)(5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)

(1 4 5 2 3 6)(7 11 8 12 9 10)(13 17 15)(14 18 16)

(1 16 10)(2 13 8)(3 18 12 5 14 11)(4 15 7 6 17 9)

(1 8)(2 10)(3 7)(4 12)(5 9)(6 11)(13 16)(14 17)(15 18)
 
G:=sub<Sym(18)| (1,2)(3,4)(5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18), (1,4,5,2,3,6)(7,11,8,12,9,10)(13,17,15)(14,18,16), (1,16,10)(2,13,8)(3,18,12,5,14,11)(4,15,7,6,17,9), (1,8)(2,10)(3,7)(4,12)(5,9)(6,11)(13,16)(14,17)(15,18)>;
 
G:=Group( (1,2)(3,4)(5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18), (1,4,5,2,3,6)(7,11,8,12,9,10)(13,17,15)(14,18,16), (1,16,10)(2,13,8)(3,18,12,5,14,11)(4,15,7,6,17,9), (1,8)(2,10)(3,7)(4,12)(5,9)(6,11)(13,16)(14,17)(15,18) );
 
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18)], [(1,4,5,2,3,6),(7,11,8,12,9,10),(13,17,15),(14,18,16)], [(1,16,10),(2,13,8),(3,18,12,5,14,11),(4,15,7,6,17,9)], [(1,8),(2,10),(3,7),(4,12),(5,9),(6,11),(13,16),(14,17),(15,18)]])
 
G:=TransitiveGroup(18,152);
 

►On 18 points - transitive group 18T153
Generators in S18
(1 2)(3 4)(5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)

(1 4 5 2 3 6)(7 12 8 10 9 11)(13 17 15)(14 18 16)

(1 13 11 2 16 8)(3 15 10 6 14 9)(4 18 7 5 17 12)

(1 11)(2 8)(3 10)(4 7)(5 12)(6 9)
 
G:=sub<Sym(18)| (1,2)(3,4)(5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18), (1,4,5,2,3,6)(7,12,8,10,9,11)(13,17,15)(14,18,16), (1,13,11,2,16,8)(3,15,10,6,14,9)(4,18,7,5,17,12), (1,11)(2,8)(3,10)(4,7)(5,12)(6,9)>;
 
G:=Group( (1,2)(3,4)(5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18), (1,4,5,2,3,6)(7,12,8,10,9,11)(13,17,15)(14,18,16), (1,13,11,2,16,8)(3,15,10,6,14,9)(4,18,7,5,17,12), (1,11)(2,8)(3,10)(4,7)(5,12)(6,9) );
 
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18)], [(1,4,5,2,3,6),(7,12,8,10,9,11),(13,17,15),(14,18,16)], [(1,13,11,2,16,8),(3,15,10,6,14,9),(4,18,7,5,17,12)], [(1,11),(2,8),(3,10),(4,7),(5,12),(6,9)]])
 
G:=TransitiveGroup(18,153);
 

►On 18 points - transitive group 18T154
Generators in S18
(1 2)(3 4)(5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)

(1 4 5 2 3 6)(7 12 8 10 9 11)(13 17 15)(14 18 16)

(1 17 9)(2 14 12)(3 13 8 5 15 7)(4 16 11 6 18 10)

(1 9)(2 12)(3 8)(4 11)(5 7)(6 10)
 
G:=sub<Sym(18)| (1,2)(3,4)(5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18), (1,4,5,2,3,6)(7,12,8,10,9,11)(13,17,15)(14,18,16), (1,17,9)(2,14,12)(3,13,8,5,15,7)(4,16,11,6,18,10), (1,9)(2,12)(3,8)(4,11)(5,7)(6,10)>;
 
G:=Group( (1,2)(3,4)(5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18), (1,4,5,2,3,6)(7,12,8,10,9,11)(13,17,15)(14,18,16), (1,17,9)(2,14,12)(3,13,8,5,15,7)(4,16,11,6,18,10), (1,9)(2,12)(3,8)(4,11)(5,7)(6,10) );
 
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18)], [(1,4,5,2,3,6),(7,12,8,10,9,11),(13,17,15),(14,18,16)], [(1,17,9),(2,14,12),(3,13,8,5,15,7),(4,16,11,6,18,10)], [(1,9),(2,12),(3,8),(4,11),(5,7),(6,10)]])
 
G:=TransitiveGroup(18,154);
 

►On 18 points - transitive group 18T155
Generators in S18
(1 2)(3 4)(5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)

(1 4 5 2 3 6)(7 10 8 11 9 12)(13 17 15)(14 18 16)

(1 15 11 2 18 7)(3 17 10 6 16 8)(4 14 9 5 13 12)

(1 7)(2 11)(3 9)(4 10)(5 8)(6 12)(13 16)(14 17)(15 18)
 
G:=sub<Sym(18)| (1,2)(3,4)(5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18), (1,4,5,2,3,6)(7,10,8,11,9,12)(13,17,15)(14,18,16), (1,15,11,2,18,7)(3,17,10,6,16,8)(4,14,9,5,13,12), (1,7)(2,11)(3,9)(4,10)(5,8)(6,12)(13,16)(14,17)(15,18)>;
 
G:=Group( (1,2)(3,4)(5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18), (1,4,5,2,3,6)(7,10,8,11,9,12)(13,17,15)(14,18,16), (1,15,11,2,18,7)(3,17,10,6,16,8)(4,14,9,5,13,12), (1,7)(2,11)(3,9)(4,10)(5,8)(6,12)(13,16)(14,17)(15,18) );
 
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8,9),(10,11,12),(13,14,15,16,17,18)], [(1,4,5,2,3,6),(7,10,8,11,9,12),(13,17,15),(14,18,16)], [(1,15,11,2,18,7),(3,17,10,6,16,8),(4,14,9,5,13,12)], [(1,7),(2,11),(3,9),(4,10),(5,8),(6,12),(13,16),(14,17),(15,18)]])
 
G:=TransitiveGroup(18,155);
 

Matrix representation of C62:5D6 ►in GL6(Z)

-100000
0-10000
001100
00-1000
000001
0000-1-1
,
110000
-100000
00-1-100
001000
000011
0000-10
,
000010
0000-1-1
100000
-1-10000
001000
00-1-100
,
000010
000001
001000
000100
100000
010000

G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1],[1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,1,-1,0,0,0,0,1,0],[0,0,1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,-1,1,-1,0,0,0,0,0,-1,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0,0,0] >;
 

C62:5D6 in GAP, Magma, Sage, TeX 

C_6^2\rtimes_5D_6
 
% in TeX
 
G:=Group("C6^2:5D6");
 
// GroupNames label
 
G:=SmallGroup(432,523);
 
// by ID
 
G=gap.SmallGroup(432,523);
 
# by ID
 
G:=PCGroup([7,-2,-2,-3,-3,-3,-2,2,93,675,353,2524,1271,4548,2287,2659,3989]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^6=c^6=d^2=1,a*b=b*a,c*a*c^-1=a^2*b^-1,d*a*d=a^-1*b^-1,c*b*c^-1=a^3*b^-1,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Character table of C62:5D6 in TeX

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