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G = C62:D4  order 288 = 25·32

2nd semidirect product of C62 and D4 acting faithfully

non-abelian, soluble, monomial

Aliases: C62:2D4, C32:C4:2D4, C22:1S3wrC2, Dic3:D6:2C2, C32:2(C4:D4), C6.D6.4C22, S32:C4:3C2, (C2xC3:S3):7D4, (C2xS3wrC2):4C2, C3:S3.7(C2xD4), C3:S3.Q8:3C2, C2.23(C2xS3wrC2), (C2xS32).4C22, (C3xC6).23(C2xD4), C3:S3.8(C4oD4), (C22xC32:C4):2C2, (C2xC3:S3).11C23, (C2xC32:C4).17C22, (C22xC3:S3).53C22, SmallGroup(288,890)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C2xC3:S3 — C62:D4
Chief series
C1 — C32 — C3:S3 — C2xC3:S3 — C2xS32 — C2xS3wrC2 — C62:D4
Lower central
C32 — C2xC3:S3 — C62:D4
Upper central
C1 — C2 — C22

Generators and relations for C62:D4
 G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, cac-1=a3b4, dad=a-1b3, cbc-1=a2b3, bd=db, dcd=c-1 >

Subgroups: 920 in 148 conjugacy classes, 27 normal (17 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2xC4, D4, C23, C32, Dic3, C12, D6, C2xC6, C22:C4, C4:C4, C22xC4, C2xD4, C3xS3, C3:S3, C3:S3, C3xC6, C3xC6, C4xS3, D12, C3:D4, C3xD4, C22xS3, C4:D4, C3xDic3, C32:C4, C32:C4, S32, S3xC6, C2xC3:S3, C2xC3:S3, C62, S3xD4, C6.D6, C3:D12, C3xC3:D4, S3wrC2, C2xC32:C4, C2xC32:C4, C2xS32, C22xC3:S3, S32:C4, C3:S3.Q8, Dic3:D6, C2xS3wrC2, C22xC32:C4, C62:D4
Quotients: C1, C2, C22, D4, C23, C2xD4, C4oD4, C4:D4, S3wrC2, C2xS3wrC2, C62:D4

Character table of C62:D4

 class 12A2B2C2D2E2F2G3A3B4A4B4C4D4E4F6A6B6C6D6E6F12A12B
 size 1129912121844121218181818448824242424
ρ1111111111111111111111111    trivial
ρ211-111-1-1-11111-11-1111-1-1-1-111    linear of order 2
ρ311-111-11-1111-11-11-111-1-11-1-11    linear of order 2
ρ4111111-11111-1-1-1-1-11111-11-11    linear of order 2
ρ511-11111-111-1-1-11-1111-1-111-1-1    linear of order 2
ρ611111-1-1111-1-111111111-1-1-1-1    linear of order 2
ρ711111-11111-11-1-1-1-111111-11-1    linear of order 2
ρ811-1111-1-111-111-11-111-1-1-111-1    linear of order 2
ρ9222-2-200-22200000022220000    orthogonal lifted from D4
ρ102-20-2200022000-202-2-2000000    orthogonal lifted from D4
ρ112-20-220002200020-2-2-2000000    orthogonal lifted from D4
ρ1222-2-2-20022200000022-2-20000    orthogonal lifted from D4
ρ132-202-200022002i0-2i0-2-2000000    complex lifted from C4oD4
ρ142-202-20002200-2i02i0-2-2000000    complex lifted from C4oD4
ρ15444000201-2020000-21-21-10-10    orthogonal lifted from S3wrC2
ρ1644-4000201-20-20000-212-1-1010    orthogonal lifted from C2xS3wrC2
ρ1744400-200-21-2000001-21-20101    orthogonal lifted from S3wrC2
ρ1844400200-212000001-21-20-10-1    orthogonal lifted from S3wrC2
ρ1944-400-200-212000001-2-12010-1    orthogonal lifted from C2xS3wrC2
ρ2044-400200-21-2000001-2-120-101    orthogonal lifted from C2xS3wrC2
ρ21444000-201-20-20000-21-211010    orthogonal lifted from S3wrC2
ρ2244-4000-201-2020000-212-110-10    orthogonal lifted from C2xS3wrC2
ρ238-8000000-42000000-24000000    orthogonal faithful
ρ248-80000002-40000004-2000000    orthogonal faithful

Permutation representations of C62:D4
►On 24 points - transitive group 24T592
Generators in S24
(7 8)(9 10)(11 12)(13 14 15)(16 17 18)(19 20 21 22 23 24)

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

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

(1 10)(2 7)(3 12)(4 9)(5 8)(6 11)(13 22)(14 24)(15 20)(16 23)(17 19)(18 21)
 
G:=sub<Sym(24)| (7,8)(9,10)(11,12)(13,14,15)(16,17,18)(19,20,21,22,23,24), (1,5,3,4,2,6)(7,11,10,8,12,9)(13,17)(14,18)(15,16)(19,22)(20,23)(21,24), (1,15)(2,13,3,14)(4,16)(5,17,6,18)(7,24,12,22)(8,21,11,19)(9,23)(10,20), (1,10)(2,7)(3,12)(4,9)(5,8)(6,11)(13,22)(14,24)(15,20)(16,23)(17,19)(18,21)>;
 
G:=Group( (7,8)(9,10)(11,12)(13,14,15)(16,17,18)(19,20,21,22,23,24), (1,5,3,4,2,6)(7,11,10,8,12,9)(13,17)(14,18)(15,16)(19,22)(20,23)(21,24), (1,15)(2,13,3,14)(4,16)(5,17,6,18)(7,24,12,22)(8,21,11,19)(9,23)(10,20), (1,10)(2,7)(3,12)(4,9)(5,8)(6,11)(13,22)(14,24)(15,20)(16,23)(17,19)(18,21) );
 
G=PermutationGroup([[(7,8),(9,10),(11,12),(13,14,15),(16,17,18),(19,20,21,22,23,24)], [(1,5,3,4,2,6),(7,11,10,8,12,9),(13,17),(14,18),(15,16),(19,22),(20,23),(21,24)], [(1,15),(2,13,3,14),(4,16),(5,17,6,18),(7,24,12,22),(8,21,11,19),(9,23),(10,20)], [(1,10),(2,7),(3,12),(4,9),(5,8),(6,11),(13,22),(14,24),(15,20),(16,23),(17,19),(18,21)]])
 
G:=TransitiveGroup(24,592);
 

►On 24 points - transitive group 24T593
Generators in S24
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)

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

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

(1 21)(2 23)(3 19)(4 24)(5 20)(6 22)(7 13)(8 15)(9 17)(10 18)(11 14)(12 16)
 
G:=sub<Sym(24)| (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,5,3,4,2,6)(7,11,8,12,9,10)(13,14,15,16,17,18)(19,24,23,22,21,20), (1,7,4,12)(2,8,6,11)(3,9,5,10)(13,21,16,24)(14,22,15,23)(17,19,18,20), (1,21)(2,23)(3,19)(4,24)(5,20)(6,22)(7,13)(8,15)(9,17)(10,18)(11,14)(12,16)>;
 
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), (1,5,3,4,2,6)(7,11,8,12,9,10)(13,14,15,16,17,18)(19,24,23,22,21,20), (1,7,4,12)(2,8,6,11)(3,9,5,10)(13,21,16,24)(14,22,15,23)(17,19,18,20), (1,21)(2,23)(3,19)(4,24)(5,20)(6,22)(7,13)(8,15)(9,17)(10,18)(11,14)(12,16) );
 
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)], [(1,5,3,4,2,6),(7,11,8,12,9,10),(13,14,15,16,17,18),(19,24,23,22,21,20)], [(1,7,4,12),(2,8,6,11),(3,9,5,10),(13,21,16,24),(14,22,15,23),(17,19,18,20)], [(1,21),(2,23),(3,19),(4,24),(5,20),(6,22),(7,13),(8,15),(9,17),(10,18),(11,14),(12,16)]])
 
G:=TransitiveGroup(24,593);
 

►On 24 points - transitive group 24T640
Generators in S24
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)

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

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

(1 6)(4 11)(7 9)(14 23)(15 17)(16 21)(18 19)(20 22)
 
G:=sub<Sym(24)| (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,4,9,6,11,7)(2,3,10,5,12,8)(13,24)(14,19)(15,20)(16,21)(17,22)(18,23), (1,24,6,13)(2,21,5,16)(3,14,10,23)(4,17,9,20)(7,15,11,22)(8,18,12,19), (1,6)(4,11)(7,9)(14,23)(15,17)(16,21)(18,19)(20,22)>;
 
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), (1,4,9,6,11,7)(2,3,10,5,12,8)(13,24)(14,19)(15,20)(16,21)(17,22)(18,23), (1,24,6,13)(2,21,5,16)(3,14,10,23)(4,17,9,20)(7,15,11,22)(8,18,12,19), (1,6)(4,11)(7,9)(14,23)(15,17)(16,21)(18,19)(20,22) );
 
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)], [(1,4,9,6,11,7),(2,3,10,5,12,8),(13,24),(14,19),(15,20),(16,21),(17,22),(18,23)], [(1,24,6,13),(2,21,5,16),(3,14,10,23),(4,17,9,20),(7,15,11,22),(8,18,12,19)], [(1,6),(4,11),(7,9),(14,23),(15,17),(16,21),(18,19),(20,22)]])
 
G:=TransitiveGroup(24,640);
 

►On 24 points - transitive group 24T647
Generators in S24
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)

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

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

(2 12)(4 9)(6 8)(13 15)(14 19)(16 23)(18 21)(20 24)
 
G:=sub<Sym(24)| (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,7,3,11,5,10)(2,8,4,12,6,9)(13,24)(14,19)(15,20)(16,21)(17,22)(18,23), (1,17)(2,14)(3,13,5,15)(4,16,6,18)(7,20,10,24)(8,23,9,21)(11,22)(12,19), (2,12)(4,9)(6,8)(13,15)(14,19)(16,23)(18,21)(20,24)>;
 
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), (1,7,3,11,5,10)(2,8,4,12,6,9)(13,24)(14,19)(15,20)(16,21)(17,22)(18,23), (1,17)(2,14)(3,13,5,15)(4,16,6,18)(7,20,10,24)(8,23,9,21)(11,22)(12,19), (2,12)(4,9)(6,8)(13,15)(14,19)(16,23)(18,21)(20,24) );
 
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)], [(1,7,3,11,5,10),(2,8,4,12,6,9),(13,24),(14,19),(15,20),(16,21),(17,22),(18,23)], [(1,17),(2,14),(3,13,5,15),(4,16,6,18),(7,20,10,24),(8,23,9,21),(11,22),(12,19)], [(2,12),(4,9),(6,8),(13,15),(14,19),(16,23),(18,21),(20,24)]])
 
G:=TransitiveGroup(24,647);
 

Matrix representation of C62:D4 ►in GL6(Z)

-100000
010000
000-100
001-100
0000-11
0000-10
,
-100000
0-10000
000-100
001-100
00000-1
00001-1
,
100000
010000
000010
000001
000100
001000
,
0-10000
-100000
000010
000001
001000
000100

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

C62:D4 in GAP, Magma, Sage, TeX 

C_6^2\rtimes D_4
 
% in TeX
 
G:=Group("C6^2:D4");
 
// GroupNames label
 
G:=SmallGroup(288,890);
 
// by ID
 
G=gap.SmallGroup(288,890);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,422,219,2693,2028,362,797,1203]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^6=c^4=d^2=1,a*b=b*a,c*a*c^-1=a^3*b^4,d*a*d=a^-1*b^3,c*b*c^-1=a^2*b^3,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

Export

Character table of C62:D4 in TeX

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