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G = C62⋊Q8order 288 = 25·32

1st semidirect product of C62 and Q8 acting faithfully

non-abelian, soluble, monomial

Aliases: C621Q8, C22⋊PSU3(𝔽2), C32⋊C4.7D4, C62⋊C4.C2, C322(C22⋊Q8), (C2×PSU3(𝔽2))⋊1C2, C2.PSU3(𝔽2)⋊2C2, C2.5(C2×PSU3(𝔽2)), (C2×C3⋊S3)⋊3Q8, C3⋊S3.9(C2×D4), (C3×C6).5(C2×Q8), C3⋊S3.9(C4○D4), (C2×C3⋊S3).16C23, (C22×C32⋊C4).7C2, (C2×C32⋊C4).21C22, (C22×C3⋊S3).55C22, SmallGroup(288,895)

Series: Derived Chief Lower central Upper central

C1C32C2×C3⋊S3 — C62⋊Q8
C1C32C3⋊S3C2×C3⋊S3C2×C32⋊C4C2×PSU3(𝔽2) — C62⋊Q8
C32C2×C3⋊S3 — C62⋊Q8
C1C2C22

Generators and relations for C62⋊Q8
 G = < a,b,c,d | a6=b6=c4=1, d2=c2, ab=ba, cac-1=a3b-1, dad-1=a-1b4, cbc-1=a4b3, dbd-1=a4b, dcd-1=c-1 >

Subgroups: 612 in 90 conjugacy classes, 27 normal (17 characteristic)
C1, C2, C2 [×4], C3, C4 [×7], C22, C22 [×4], S3 [×4], C6 [×3], C2×C4 [×8], Q8 [×2], C23, C32, D6 [×6], C2×C6, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, C3⋊S3 [×2], C3⋊S3, C3×C6, C3×C6, C22×S3, C22⋊Q8, C32⋊C4 [×2], C32⋊C4 [×5], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, PSU3(𝔽2) [×2], C2×C32⋊C4 [×2], C2×C32⋊C4 [×4], C2×C32⋊C4 [×2], C22×C3⋊S3, C2.PSU3(𝔽2), C2.PSU3(𝔽2) [×2], C62⋊C4 [×2], C2×PSU3(𝔽2), C22×C32⋊C4, C62⋊Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, C2×D4, C2×Q8, C4○D4, C22⋊Q8, PSU3(𝔽2), C2×PSU3(𝔽2), C62⋊Q8

Character table of C62⋊Q8

 class 12A2B2C2D2E34A4B4C4D4E4F4G4H6A6B6C
 size 112991881818181836363636888
ρ1111111111111111111    trivial
ρ21111111-1-1-1-1-1-111111    linear of order 2
ρ311-111-11-1-111-11-11-1-11    linear of order 2
ρ411-111-11-1-1111-11-1-1-11    linear of order 2
ρ51111111-1-1-1-111-1-1111    linear of order 2
ρ611-111-1111-1-11-1-11-1-11    linear of order 2
ρ711-111-1111-1-1-111-1-1-11    linear of order 2
ρ811111111111-1-1-1-1111    linear of order 2
ρ92-20-22022-200000000-2    orthogonal lifted from D4
ρ102-20-2202-2200000000-2    orthogonal lifted from D4
ρ11222-2-2-2200000000222    symplectic lifted from Q8, Schur index 2
ρ1222-2-2-22200000000-2-22    symplectic lifted from Q8, Schur index 2
ρ132-202-20200-2i2i000000-2    complex lifted from C4○D4
ρ142-202-202002i-2i000000-2    complex lifted from C4○D4
ρ15888000-100000000-1-1-1    orthogonal lifted from PSU3(𝔽2)
ρ168-80000-100000000-331    orthogonal faithful
ρ1788-8000-10000000011-1    orthogonal lifted from C2×PSU3(𝔽2)
ρ188-80000-1000000003-31    orthogonal faithful

Permutation representations of C62⋊Q8
On 24 points - transitive group 24T632
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 5 2 3 6)(7 15 8 13 9 14)(10 16)(11 17)(12 18)(19 20 21 22 23 24)
(1 12 5 10)(2 18 6 16)(3 11)(4 17)(7 19)(8 23 9 21)(13 22)(14 20 15 24)
(1 21 5 23)(2 24 6 20)(3 19)(4 22)(7 11)(8 12 9 10)(13 17)(14 18 15 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), (1,4,5,2,3,6)(7,15,8,13,9,14)(10,16)(11,17)(12,18)(19,20,21,22,23,24), (1,12,5,10)(2,18,6,16)(3,11)(4,17)(7,19)(8,23,9,21)(13,22)(14,20,15,24), (1,21,5,23)(2,24,6,20)(3,19)(4,22)(7,11)(8,12,9,10)(13,17)(14,18,15,16)>;

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

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

G:=TransitiveGroup(24,632);

On 24 points - transitive group 24T633
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 9 12 8 11)(13 18 17 16 15 14)(19 22)(20 23)(21 24)
(1 20 4 23)(2 22 6 21)(3 24 5 19)(7 13 10 18)(8 15 12 16)(9 17 11 14)
(1 11 4 9)(2 12 6 8)(3 10 5 7)(13 19 18 24)(14 20 17 23)(15 21 16 22)

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

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

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

G:=TransitiveGroup(24,633);

Matrix representation of C62⋊Q8 in GL8(ℤ)

-1-1000000
10000000
00-100000
000-10000
0000-1-100
00001000
00000011
000000-10
,
-10000000
0-1000000
000-10000
00110000
00000-100
00001100
00000011
000000-10
,
00100000
00010000
-10000000
11000000
000000-10
00000011
00001000
00000100
,
00001000
00000100
00000010
00000001
-10000000
11000000
00-100000
00110000

G:=sub<GL(8,Integers())| [-1,1,0,0,0,0,0,0,-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,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,1,0],[-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,-1,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,1,0],[0,0,-1,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,1,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0] >;

C62⋊Q8 in GAP, Magma, Sage, TeX

C_6^2\rtimes Q_8
% in TeX

G:=Group("C6^2:Q8");
// GroupNames label

G:=SmallGroup(288,895);
// by ID

G=gap.SmallGroup(288,895);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,141,64,219,9413,2028,362,12550,1581,1203]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=c^4=1,d^2=c^2,a*b=b*a,c*a*c^-1=a^3*b^-1,d*a*d^-1=a^-1*b^4,c*b*c^-1=a^4*b^3,d*b*d^-1=a^4*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Character table of C62⋊Q8 in TeX

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