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G = C62.9D4order 288 = 25·32

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

non-abelian, soluble, monomial

Aliases: C62.9D4, C22.4S3≀C2, C62⋊C44C2, Dic3⋊D6.2C2, C32⋊(C22.D4), C6.D6.13C22, S32⋊C45C2, C3⋊S3.Q82C2, 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

C1C32C2×C3⋊S3 — C62.9D4
C1C32C3⋊S3C2×C3⋊S3C2×S32S32⋊C4 — C62.9D4
C32C2×C3⋊S3 — C62.9D4
C1C2C22

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 [×5], C3 [×2], C4 [×5], C22, C22 [×7], S3 [×8], C6 [×6], C2×C4 [×7], D4 [×2], C23 [×2], C32, Dic3 [×3], C12 [×3], D6 [×13], C2×C6 [×3], C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, C3×S3, C3⋊S3 [×2], C3⋊S3, C3×C6, C3×C6, C4×S3 [×5], D12, C2×Dic3, C3⋊D4 [×2], C2×C12, C3×D4, C22×S3 [×3], C22.D4, C3×Dic3 [×3], C32⋊C4 [×2], S32 [×2], S3×C6, C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, S3×C2×C4, S3×D4, C6.D6, C6.D6 [×2], C6.D6, C3⋊D12, C6×Dic3, C3×C3⋊D4, C2×C32⋊C4 [×2], C2×S32, C22×C3⋊S3, S32⋊C4 [×2], C3⋊S3.Q8 [×2], C62⋊C4, C2×C6.D6, Dic3⋊D6, C62.9D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, C4○D4 [×2], 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 9 12 8 10 11)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)
(1 21 4 24)(2 19 6 20)(3 23 5 22)(7 15 8 18)(9 16 12 14)(10 13 11 17)
(1 7)(2 10)(3 12)(4 8)(5 9)(6 11)(13 23)(14 19)(15 21)(16 20)(17 22)(18 24)

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

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

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

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

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

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

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

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