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G = C3⋊S3.Q8order 144 = 24·32

The non-split extension by C3⋊S3 of Q8 acting via Q8/C2=C22

non-abelian, soluble, monomial

Aliases: C3⋊S3.Q8, C2.2S3≀C2, C32⋊C41C4, (C3×C6).2D4, C321(C4⋊C4), C6.D6.2C2, C3⋊S3.3(C2×C4), (C2×C32⋊C4).1C2, (C2×C3⋊S3).2C22, SmallGroup(144,116)

Series: Derived Chief Lower central Upper central

C1C32C3⋊S3 — C3⋊S3.Q8
C1C32C3⋊S3C2×C3⋊S3C6.D6 — C3⋊S3.Q8
C32C3⋊S3 — C3⋊S3.Q8
C1C2

Generators and relations for C3⋊S3.Q8
 G = < a,b,c,d,e | a3=b3=c2=d4=1, e2=d2, ab=ba, cac=dbd-1=a-1, dad-1=cbc=ebe-1=b-1, ae=ea, cd=dc, ce=ec, ede-1=cd-1 >

9C2
9C2
2C3
2C3
6C4
6C4
9C4
9C22
9C4
2C6
2C6
6S3
6S3
6S3
6S3
9C2×C4
9C2×C4
9C2×C4
2Dic3
2Dic3
6C12
6C12
6D6
6D6
9C4⋊C4
6C4×S3
6C4×S3
2C3×Dic3
2C3×Dic3

Character table of C3⋊S3.Q8

 class 12A2B2C3A3B4A4B4C4D4E4F6A6B12A12B12C12D
 size 119944666618184412121212
ρ1111111111111111111    trivial
ρ2111111-11-11-1-1111-1-11    linear of order 2
ρ31111111-11-1-1-111-111-1    linear of order 2
ρ4111111-1-1-1-11111-1-1-1-1    linear of order 2
ρ51-1-1111-iii-i-11-1-1i-ii-i    linear of order 4
ρ61-1-1111ii-i-i1-1-1-1ii-i-i    linear of order 4
ρ71-1-1111-i-iii1-1-1-1-i-iii    linear of order 4
ρ81-1-1111i-i-ii-11-1-1-ii-ii    linear of order 4
ρ922-2-222000000220000    orthogonal lifted from D4
ρ102-22-222000000-2-20000    symplectic lifted from Q8, Schur index 2
ρ1144001-2020200-21-100-1    orthogonal lifted from S3≀C2
ρ124400-212020001-20-1-10    orthogonal lifted from S3≀C2
ρ134400-21-20-20001-20110    orthogonal lifted from S3≀C2
ρ1444001-20-20-200-211001    orthogonal lifted from S3≀C2
ρ154-400-21-2i02i000-120i-i0    complex faithful
ρ164-4001-20-2i02i002-1i00-i    complex faithful
ρ174-4001-202i0-2i002-1-i00i    complex faithful
ρ184-400-212i0-2i000-120-ii0    complex faithful

Permutation representations of C3⋊S3.Q8
On 24 points - transitive group 24T214
Generators in S24
(1 20 15)(3 18 13)(6 11 24)(8 9 22)
(2 16 17)(4 14 19)(5 23 10)(7 21 12)
(1 3)(2 4)(5 12)(6 9)(7 10)(8 11)(13 20)(14 17)(15 18)(16 19)(21 23)(22 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 21 4 23)(5 16 7 14)(6 18 8 20)(9 15 11 13)(10 17 12 19)

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

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

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

G:=TransitiveGroup(24,214);

On 24 points - transitive group 24T216
Generators in S24
(1 13 18)(3 15 20)(6 11 22)(8 9 24)
(2 19 14)(4 17 16)(5 21 10)(7 23 12)
(5 10)(6 11)(7 12)(8 9)(13 18)(14 19)(15 20)(16 17)
(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 19 7 17)(6 13 8 15)(9 20 11 18)(10 14 12 16)

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

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

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

G:=TransitiveGroup(24,216);

C3⋊S3.Q8 is a maximal subgroup of
PSU3(𝔽2)⋊C4  F9⋊C4  S32⋊Q8  C32⋊C4⋊Q8  C4×S3≀C2  C62.9D4  C62⋊D4  C33⋊C4⋊C4  (C3×C6).9D12
C3⋊S3.Q8 is a maximal quotient of
C32⋊C4⋊C8  C4.19S3≀C2  C62.D4  C62.6D4  C62.7D4  C6.S3≀C2  C33⋊C4⋊C4  (C3×C6).9D12

Matrix representation of C3⋊S3.Q8 in GL4(𝔽5) generated by

4010
0034
4000
2134
,
4200
2000
0203
2134
,
4002
0021
0302
0001
,
3004
0302
0034
0002
,
2100
0300
0104
0310
G:=sub<GL(4,GF(5))| [4,0,4,2,0,0,0,1,1,3,0,3,0,4,0,4],[4,2,0,2,2,0,2,1,0,0,0,3,0,0,3,4],[4,0,0,0,0,0,3,0,0,2,0,0,2,1,2,1],[3,0,0,0,0,3,0,0,0,0,3,0,4,2,4,2],[2,0,0,0,1,3,1,3,0,0,0,1,0,0,4,0] >;

C3⋊S3.Q8 in GAP, Magma, Sage, TeX

C_3\rtimes S_3.Q_8
% in TeX

G:=Group("C3:S3.Q8");
// GroupNames label

G:=SmallGroup(144,116);
// by ID

G=gap.SmallGroup(144,116);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,3,48,73,55,964,730,256,299,881]);
// Polycyclic

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

Export

Subgroup lattice of C3⋊S3.Q8 in TeX
Character table of C3⋊S3.Q8 in TeX

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