Copied to
clipboard

G = C62.9D4  order 288 = 25·32

9th non-split extension by C62 of D4 acting faithfully

non-abelian, soluble, monomial

Aliases: C62.9D4, C22.4S3≀C2, C62⋊C4⋊4C2, Dic3⋊D6.2C2, C32⋊(C22.D4), C6.D6.13C22, S32⋊C4⋊5C2, C3⋊S3.Q8⋊2C2, C2.15(C2×S3≀C2), (C2×C3⋊S3).35D4, (C2×S32).3C22, (C3×C6).14(C2×D4), C3⋊S3.7(C4○D4), (C2×C3⋊S3).8C23, (C2×C6.D6)⋊17C2, (C2×C32⋊C4).6C22, (C22×C3⋊S3).50C22, SmallGroup(288,881)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C2×C3⋊S3 — C62.9D4
Chief series
C1 — C32 — C3⋊S3 — C2×C3⋊S3 — C2×S32 — S32⋊C4 — C62.9D4
Lower central
C32 — C2×C3⋊S3 — C62.9D4
Upper central
C1 — C2 — C22

Generators and relations for C62.9D4
 G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, cac-1=a3b, dad=a-1b3, cbc-1=a2b3, bd=db, dcd=b3c-1 >

Subgroups: 744 in 132 conjugacy classes, 25 normal (15 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, D4, C23, C32, Dic3, C12, D6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×S3, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22.D4, C3×Dic3, C32⋊C4, S32, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, S3×C2×C4, S3×D4, C6.D6, C6.D6, C6.D6, C3⋊D12, C6×Dic3, C3×C3⋊D4, C2×C32⋊C4, C2×S32, C22×C3⋊S3, S32⋊C4, C3⋊S3.Q8, C62⋊C4, C2×C6.D6, Dic3⋊D6, C62.9D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C22.D4, S3≀C2, C2×S3≀C2, C62.9D4

Character table of C62.9D4

 class 12A2B2C2D2E2F3A3B4A4B4C4D4E4F4G6A6B6C6D6E6F12A12B12C12D12E
 size 11299121844666612363644448241212121224
ρ1111111111111111111111111111    trivial
ρ2111111111-1-1-1-11-1-1111111-1-1-1-11    linear of order 2
ρ311-1111-1111-1-11-1-111-1-11-111-1-11-1    linear of order 2
ρ411-1111-111-111-1-11-11-1-11-11-111-1-1    linear of order 2
ρ511111-1111-1-1-1-1-11111111-1-1-1-1-1-1    linear of order 2
ρ611111-11111111-1-1-111111-11111-1    linear of order 2
ρ711-111-1-1111-1-1111-11-1-11-1-11-1-111    linear of order 2
ρ811-111-1-111-111-11-111-1-11-1-1-111-11    linear of order 2
ρ9222-2-20-222000000022222000000    orthogonal lifted from D4
ρ1022-2-2-2022200000002-2-22-2000000    orthogonal lifted from D4
ρ112-20-22002202i-2i0000-200-2000-2i2i00    complex lifted from C4○D4
ρ122-20-2200220-2i2i0000-200-20002i-2i00    complex lifted from C4○D4
ρ132-202-200222i00-2i000-200-2002i00-2i0    complex lifted from C4○D4
ρ142-202-20022-2i002i000-200-200-2i002i0    complex lifted from C4○D4
ρ1544-400-20-210000200122-2-110000-1    orthogonal lifted from C2×S3≀C2
ρ1644-400001-22-2-22000-2-1-1120-111-10    orthogonal lifted from C2×S3≀C2
ρ1744-400001-2-222-2000-2-1-11201-1-110    orthogonal lifted from C2×S3≀C2
ρ184440020-2100002001-2-2-21-10000-1    orthogonal lifted from S3≀C2
ρ1944400001-22222000-2111-20-1-1-1-10    orthogonal lifted from S3≀C2
ρ2044400001-2-2-2-2-2000-2111-2011110    orthogonal lifted from S3≀C2
ρ2144-40020-210000-200122-2-1-100001    orthogonal lifted from C2×S3≀C2
ρ2244400-20-210000-2001-2-2-21100001    orthogonal lifted from S3≀C2
ρ234-4000001-2-2i2i-2i2i00023-3-100ii-i-i0    complex faithful
ρ244-4000001-2-2i-2i2i2i0002-33-100i-ii-i0    complex faithful
ρ254-4000001-22i2i-2i-2i0002-33-100-ii-ii0    complex faithful
ρ264-4000001-22i-2i2i-2i00023-3-100-i-iii0    complex faithful
ρ278-800000-420000000-20040000000    orthogonal faithful

Permutation representations of C62.9D4
►On 24 points - transitive group 24T595
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 10 11 8 9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)

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

(1 11)(2 7)(3 9)(4 12)(5 8)(6 10)(13 24)(14 20)(15 22)(16 21)(17 23)(18 19)
 
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)(7,10,11,8,9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,22,4,19)(2,20,6,21)(3,24,5,23)(7,13,10,17)(8,16,9,14)(11,15,12,18), (1,11)(2,7)(3,9)(4,12)(5,8)(6,10)(13,24)(14,20)(15,22)(16,21)(17,23)(18,19)>;
 
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)(7,10,11,8,9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24), (1,22,4,19)(2,20,6,21)(3,24,5,23)(7,13,10,17)(8,16,9,14)(11,15,12,18), (1,11)(2,7)(3,9)(4,12)(5,8)(6,10)(13,24)(14,20)(15,22)(16,21)(17,23)(18,19) );
 
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),(7,10,11,8,9,12),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24)], [(1,22,4,19),(2,20,6,21),(3,24,5,23),(7,13,10,17),(8,16,9,14),(11,15,12,18)], [(1,11),(2,7),(3,9),(4,12),(5,8),(6,10),(13,24),(14,20),(15,22),(16,21),(17,23),(18,19)]])
 
G:=TransitiveGroup(24,595);
 

►On 24 points - transitive group 24T599
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 9 10 8 12)(13 18 17 16 15 14)(19 24 23 22 21 20)

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

(1 13)(2 15)(3 17)(4 16)(5 18)(6 14)(7 19)(8 21)(9 23)(10 22)(11 24)(12 20)
 
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)(7,11,9,10,8,12)(13,18,17,16,15,14)(19,24,23,22,21,20), (1,22,4,19)(2,20,6,21)(3,24,5,23)(7,13,10,16)(8,17,12,18)(9,15,11,14), (1,13)(2,15)(3,17)(4,16)(5,18)(6,14)(7,19)(8,21)(9,23)(10,22)(11,24)(12,20)>;
 
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)(7,11,9,10,8,12)(13,18,17,16,15,14)(19,24,23,22,21,20), (1,22,4,19)(2,20,6,21)(3,24,5,23)(7,13,10,16)(8,17,12,18)(9,15,11,14), (1,13)(2,15)(3,17)(4,16)(5,18)(6,14)(7,19)(8,21)(9,23)(10,22)(11,24)(12,20) );
 
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),(7,11,9,10,8,12),(13,18,17,16,15,14),(19,24,23,22,21,20)], [(1,22,4,19),(2,20,6,21),(3,24,5,23),(7,13,10,16),(8,17,12,18),(9,15,11,14)], [(1,13),(2,15),(3,17),(4,16),(5,18),(6,14),(7,19),(8,21),(9,23),(10,22),(11,24),(12,20)]])
 
G:=TransitiveGroup(24,599);
 

►On 24 points - transitive group 24T646
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 11 9 4 5 7)(2 12 10 3 6 8)(13 21)(14 22)(15 23)(16 24)(17 19)(18 20)

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

(1 4)(5 11)(7 9)(13 23)(15 21)(16 18)(17 19)(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), (1,11,9,4,5,7)(2,12,10,3,6,8)(13,21)(14,22)(15,23)(16,24)(17,19)(18,20), (1,19,4,17)(2,14,3,22)(5,23,7,13)(6,18,8,24)(9,21,11,15)(10,16,12,20), (1,4)(5,11)(7,9)(13,23)(15,21)(16,18)(17,19)(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), (1,11,9,4,5,7)(2,12,10,3,6,8)(13,21)(14,22)(15,23)(16,24)(17,19)(18,20), (1,19,4,17)(2,14,3,22)(5,23,7,13)(6,18,8,24)(9,21,11,15)(10,16,12,20), (1,4)(5,11)(7,9)(13,23)(15,21)(16,18)(17,19)(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)], [(1,11,9,4,5,7),(2,12,10,3,6,8),(13,21),(14,22),(15,23),(16,24),(17,19),(18,20)], [(1,19,4,17),(2,14,3,22),(5,23,7,13),(6,18,8,24),(9,21,11,15),(10,16,12,20)], [(1,4),(5,11),(7,9),(13,23),(15,21),(16,18),(17,19),(20,24)]])
 
G:=TransitiveGroup(24,646);
 

Matrix representation of C62.9D4 ►in GL4(𝔽5) generated by

0320
2320
2223
2020
,
2010
1111
2040
1400
,
2320
0113
3433
4234
,
2141
3222
0314
4410
G:=sub<GL(4,GF(5))| [0,2,2,2,3,3,2,0,2,2,2,2,0,0,3,0],[2,1,2,1,0,1,0,4,1,1,4,0,0,1,0,0],[2,0,3,4,3,1,4,2,2,1,3,3,0,3,3,4],[2,3,0,4,1,2,3,4,4,2,1,1,1,2,4,0] >;
 

C62.9D4 in GAP, Magma, Sage, TeX 

C_6^2._9D_4
 
% in TeX
 
G:=Group("C6^2.9D4");
 
// GroupNames label
 
G:=SmallGroup(288,881);
 
// by ID
 
G=gap.SmallGroup(288,881);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,141,120,422,2693,2028,362,797,1203]);
 
// 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,d*a*d=a^-1*b^3,c*b*c^-1=a^2*b^3,b*d=d*b,d*c*d=b^3*c^-1>;
 
// generators/relations
 

Export

Character table of C62.9D4 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁