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G = C62⋊Q8  order 288 = 25·32

1st semidirect product of C62 and Q8 acting faithfully

non-abelian, soluble, monomial

Aliases: C62⋊1Q8, C22⋊PSU3(𝔽2), C32⋊C4.7D4, C62⋊C4.C2, C32⋊2(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

Derived series
C1 — C32 — C2×C3⋊S3 — C62⋊Q8
Chief series
C1 — C32 — C3⋊S3 — C2×C3⋊S3 — C2×C32⋊C4 — C2×PSU3(𝔽2) — C62⋊Q8
Lower central
C32 — C2×C3⋊S3 — C62⋊Q8
Upper central
C1 — C2 — C22

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, C3, C4, C22, C22, S3, C6, C2×C4, Q8, C23, C32, D6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C22×S3, C22⋊Q8, C32⋊C4, C32⋊C4, C2×C3⋊S3, C2×C3⋊S3, C62, PSU3(𝔽2), C2×C32⋊C4, C2×C32⋊C4, C2×C32⋊C4, C22×C3⋊S3, C2.PSU3(𝔽2), C2.PSU3(𝔽2), C62⋊C4, C2×PSU3(𝔽2), C22×C32⋊C4, C62⋊Q8
Quotients: C1, C2, C22, D4, Q8, 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 6 3 2 5 4)(7 18)(8 16)(9 17)(10 15 11 13 12 14)(19 20 21 22 23 24)

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

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

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

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