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G = C33⋊Q8order 216 = 23·33

2nd semidirect product of C33 and Q8 acting faithfully

non-abelian, soluble, monomial

Aliases: C332Q8, C3⋊PSU3(𝔽2), C322Dic6, 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

C1C32C3×C3⋊S3 — C33⋊Q8
C1C3C33C3×C3⋊S3C33⋊C4 — C33⋊Q8
C33C3×C3⋊S3 — C33⋊Q8
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 >

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

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
ρ82-2-1-1-120001-33    symplectic lifted from Dic6, Schur index 2
ρ92-2-1-1-1200013-3    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 8 10)(2 21 23)(3 15 13)(4 5 12)(6 14 7)(9 24 25)(11 27 26)(16 22 19)(17 20 18)
(1 11 9)(2 20 22)(3 14 12)(4 15 7)(5 13 6)(8 27 24)(10 26 25)(16 23 17)(18 19 21)
(1 3 2)(4 19 24)(5 16 25)(6 17 26)(7 18 27)(8 15 21)(9 12 22)(10 13 23)(11 14 20)
(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 21 6 23)(5 20 7 22)(8 26 10 24)(9 25 11 27)(12 16 14 18)(13 19 15 17)

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

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

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

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

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