Copied to
clipboard

G = C33⋊Q8  order 216 = 23·33

2nd semidirect product of C33 and Q8 acting faithfully

non-abelian, soluble, monomial

Aliases: C33⋊2Q8, C3⋊PSU3(𝔽2), C32⋊2Dic6, C3⋊S3.3D6, C32⋊C4.2S3, C33⋊C4.C2, (C3×C32⋊C4).2C2, (C3×C3⋊S3).6C22, SmallGroup(216,161)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32 — C3×C3⋊S3 — C33⋊Q8
Chief series
C1 — C3 — C33 — C3×C3⋊S3 — C33⋊C4 — C33⋊Q8
Lower central
C33 — C3×C3⋊S3 — C33⋊Q8
Upper central
C1

Generators and relations for C33⋊Q8
 G = < a,b,c,d,e | a3=b3=c3=d4=1, e2=d2, ab=ba, ac=ca, dad-1=b, eae-1=ab-1, bc=cb, dbd-1=a-1, ebe-1=a-1b-1, cd=dc, ece-1=c-1, ede-1=d-1 >

C1
9C2
C3
4C3
8C3
9C4
27C4
27C4
9C6
12S3
C32
4C32
8C32
27Q8
9C12
9Dic3
9Dic3
C3⋊S3
12C3×S3
C33
9Dic6
C32⋊C4
3C32⋊C4
3C32⋊C4
C3×C3⋊S3
3PSU3(𝔽2)
C33⋊C4
C3×C32⋊C4
C33⋊C4
C33⋊Q8

Character table of C33⋊Q8

 class 123A3B3C3D4A4B4C612A12B
 size 192888185454181818
ρ1111111111111    trivial
ρ21111111-1-1111    linear of order 2
ρ3111111-1-111-1-1    linear of order 2
ρ4111111-11-11-1-1    linear of order 2
ρ522-1-1-12200-1-1-1    orthogonal lifted from S3
ρ622-1-1-12-200-111    orthogonal lifted from D6
ρ72-22222000-200    symplectic lifted from Q8, Schur index 2
ρ82-2-1-1-120001-√3√3    symplectic lifted from Dic6, Schur index 2
ρ92-2-1-1-120001√3-√3    symplectic lifted from Dic6, Schur index 2
ρ10808-1-1-1000000    orthogonal lifted from PSU3(𝔽2)
ρ1180-41+3√-3/21-3√-3/2-1000000    complex faithful
ρ1280-41-3√-3/21+3√-3/2-1000000    complex faithful

Permutation representations of C33⋊Q8
►On 24 points - transitive group 24T566
Generators in S24
(1 19 14)(3 16 17)(5 21 10)(6 22 11)(7 12 23)(8 9 24)

(2 15 20)(4 18 13)(5 21 10)(6 11 22)(7 12 23)(8 24 9)

(1 14 19)(2 15 20)(3 16 17)(4 13 18)(5 10 21)(6 11 22)(7 12 23)(8 9 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 24 3 22)(2 23 4 21)(5 20 7 18)(6 19 8 17)(9 16 11 14)(10 15 12 13)
 
G:=sub<Sym(24)| (1,19,14)(3,16,17)(5,21,10)(6,22,11)(7,12,23)(8,9,24), (2,15,20)(4,18,13)(5,21,10)(6,11,22)(7,12,23)(8,24,9), (1,14,19)(2,15,20)(3,16,17)(4,13,18)(5,10,21)(6,11,22)(7,12,23)(8,9,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,24,3,22)(2,23,4,21)(5,20,7,18)(6,19,8,17)(9,16,11,14)(10,15,12,13)>;
 
G:=Group( (1,19,14)(3,16,17)(5,21,10)(6,22,11)(7,12,23)(8,9,24), (2,15,20)(4,18,13)(5,21,10)(6,11,22)(7,12,23)(8,24,9), (1,14,19)(2,15,20)(3,16,17)(4,13,18)(5,10,21)(6,11,22)(7,12,23)(8,9,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,24,3,22)(2,23,4,21)(5,20,7,18)(6,19,8,17)(9,16,11,14)(10,15,12,13) );
 
G=PermutationGroup([[(1,19,14),(3,16,17),(5,21,10),(6,22,11),(7,12,23),(8,9,24)], [(2,15,20),(4,18,13),(5,21,10),(6,11,22),(7,12,23),(8,24,9)], [(1,14,19),(2,15,20),(3,16,17),(4,13,18),(5,10,21),(6,11,22),(7,12,23),(8,9,24)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,24,3,22),(2,23,4,21),(5,20,7,18),(6,19,8,17),(9,16,11,14),(10,15,12,13)]])
 
G:=TransitiveGroup(24,566);
 

►On 27 points - transitive group 27T87
Generators in S27
(1 19 17)(2 21 23)(3 15 13)(4 7 14)(5 6 12)(8 20 9)(10 11 22)(16 25 26)(18 24 27)

(1 18 16)(2 20 22)(3 14 12)(4 5 15)(6 13 7)(8 11 23)(9 10 21)(17 27 26)(19 24 25)

(1 3 2)(4 9 24)(5 10 25)(6 11 26)(7 8 27)(12 22 16)(13 23 17)(14 20 18)(15 21 19)

(4 5 6 7)(8 9 10 11)(12 13 14 15)(16 17 18 19)(20 21 22 23)(24 25 26 27)

(2 3)(4 22 6 20)(5 21 7 23)(8 13 10 15)(9 12 11 14)(16 26 18 24)(17 25 19 27)
 
G:=sub<Sym(27)| (1,19,17)(2,21,23)(3,15,13)(4,7,14)(5,6,12)(8,20,9)(10,11,22)(16,25,26)(18,24,27), (1,18,16)(2,20,22)(3,14,12)(4,5,15)(6,13,7)(8,11,23)(9,10,21)(17,27,26)(19,24,25), (1,3,2)(4,9,24)(5,10,25)(6,11,26)(7,8,27)(12,22,16)(13,23,17)(14,20,18)(15,21,19), (4,5,6,7)(8,9,10,11)(12,13,14,15)(16,17,18,19)(20,21,22,23)(24,25,26,27), (2,3)(4,22,6,20)(5,21,7,23)(8,13,10,15)(9,12,11,14)(16,26,18,24)(17,25,19,27)>;
 
G:=Group( (1,19,17)(2,21,23)(3,15,13)(4,7,14)(5,6,12)(8,20,9)(10,11,22)(16,25,26)(18,24,27), (1,18,16)(2,20,22)(3,14,12)(4,5,15)(6,13,7)(8,11,23)(9,10,21)(17,27,26)(19,24,25), (1,3,2)(4,9,24)(5,10,25)(6,11,26)(7,8,27)(12,22,16)(13,23,17)(14,20,18)(15,21,19), (4,5,6,7)(8,9,10,11)(12,13,14,15)(16,17,18,19)(20,21,22,23)(24,25,26,27), (2,3)(4,22,6,20)(5,21,7,23)(8,13,10,15)(9,12,11,14)(16,26,18,24)(17,25,19,27) );
 
G=PermutationGroup([[(1,19,17),(2,21,23),(3,15,13),(4,7,14),(5,6,12),(8,20,9),(10,11,22),(16,25,26),(18,24,27)], [(1,18,16),(2,20,22),(3,14,12),(4,5,15),(6,13,7),(8,11,23),(9,10,21),(17,27,26),(19,24,25)], [(1,3,2),(4,9,24),(5,10,25),(6,11,26),(7,8,27),(12,22,16),(13,23,17),(14,20,18),(15,21,19)], [(4,5,6,7),(8,9,10,11),(12,13,14,15),(16,17,18,19),(20,21,22,23),(24,25,26,27)], [(2,3),(4,22,6,20),(5,21,7,23),(8,13,10,15),(9,12,11,14),(16,26,18,24),(17,25,19,27)]])
 
G:=TransitiveGroup(27,87);
 

C33⋊Q8 is a maximal subgroup of   C33⋊SD16  F9⋊S3  S3×PSU3(𝔽2)
C33⋊Q8 is a maximal quotient of   C6.PSU3(𝔽2)  C6.2PSU3(𝔽2)

Matrix representation of C33⋊Q8 ►in GL8(𝔽13)

30000000
49000000
00100000
00010000
00109000
001000300
00000030
40000009
,
10000000
01000000
00300000
00490000
00003000
00400900
100000030
10000009
,
90000000
09000000
00900000
00090000
00903000
00900300
90000030
90000003
,
00100000
00010000
18000000
012000000
012000001
012000010
00001000
00000100
,
00108000
000012100
10000080
000000121
010000120
000000120
000112000
000012000

G:=sub<GL(8,GF(13))| [3,4,0,0,0,0,0,4,0,9,0,0,0,0,0,0,0,0,1,0,1,10,0,0,0,0,0,1,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,9],[1,0,0,0,0,0,10,1,0,1,0,0,0,0,0,0,0,0,3,4,0,4,0,0,0,0,0,9,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,9],[9,0,0,0,0,0,9,9,0,9,0,0,0,0,0,0,0,0,9,0,9,9,0,0,0,0,0,9,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3],[0,0,1,0,0,0,0,0,0,0,8,12,12,12,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,8,12,0,0,0,0,12,12,0,1,0,0,0,0,0,0,0,0,8,12,12,12,0,0,0,0,0,1,0,0,0,0] >;
 

C33⋊Q8 in GAP, Magma, Sage, TeX 

C_3^3\rtimes Q_8
 
% in TeX
 
G:=Group("C3^3:Q8");
 
// GroupNames label
 
G:=SmallGroup(216,161);
 
// by ID
 
G=gap.SmallGroup(216,161);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-3,3,-3,24,73,31,963,585,111,964,130,376,5189]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,d*a*d^-1=b,e*a*e^-1=a*b^-1,b*c=c*b,d*b*d^-1=a^-1,e*b*e^-1=a^-1*b^-1,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
 
// generators/relations
 

Export

Subgroup lattice of C33⋊Q8 in TeX
Character table of C33⋊Q8 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁