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G = C334C8order 216 = 23·33

2nd semidirect product of C33 and C8 acting via C8/C2=C4

metabelian, soluble, monomial, A-group

Aliases: C334C8, C6.(C32⋊C4), C3⋊(C322C8), C324(C3⋊C8), C2.(C33⋊C4), C3⋊Dic3.2S3, (C32×C6).2C4, (C3×C6).5Dic3, (C3×C3⋊Dic3).4C2, SmallGroup(216,118)

Series: Derived Chief Lower central Upper central

C1C33 — C334C8
C1C3C33C32×C6C3×C3⋊Dic3 — C334C8
C33 — C334C8
C1C2

Generators and relations for C334C8
 G = < a,b,c,d | a3=b3=c3=d8=1, ab=ba, ac=ca, dad-1=ab-1, bc=cb, dbd-1=a-1b-1, dcd-1=c-1 >

2C3
2C3
4C3
4C3
9C4
2C6
2C6
4C6
4C6
2C32
2C32
4C32
4C32
27C8
6Dic3
6Dic3
9C12
2C3×C6
2C3×C6
4C3×C6
4C3×C6
9C3⋊C8
6C3×Dic3
6C3×Dic3
3C322C8

Character table of C334C8

 class 123A3B3C3D3E3F3G4A4B6A6B6C6D6E6F6G8A8B8C8D12A12B
 size 112444444992444444272727271818
ρ1111111111111111111111111    trivial
ρ2111111111111111111-1-1-1-111    linear of order 2
ρ3111111111-1-11111111-iii-i-1-1    linear of order 4
ρ4111111111-1-11111111i-i-ii-1-1    linear of order 4
ρ51-11111111-ii-1-1-1-1-1-1-1ζ83ζ85ζ8ζ87-ii    linear of order 8
ρ61-11111111-ii-1-1-1-1-1-1-1ζ87ζ8ζ85ζ83-ii    linear of order 8
ρ71-11111111i-i-1-1-1-1-1-1-1ζ8ζ87ζ83ζ85i-i    linear of order 8
ρ81-11111111i-i-1-1-1-1-1-1-1ζ85ζ83ζ87ζ8i-i    linear of order 8
ρ922-12-12-1-1-122-1-12-1-1-120000-1-1    orthogonal lifted from S3
ρ1022-12-12-1-1-1-2-2-1-12-1-1-12000011    symplectic lifted from Dic3, Schur index 2
ρ112-2-12-12-1-1-12i-2i11-2111-20000-ii    complex lifted from C3⋊C8
ρ122-2-12-12-1-1-1-2i2i11-2111-20000i-i    complex lifted from C3⋊C8
ρ134441-2-211-2004-2111-2-2000000    orthogonal lifted from C32⋊C4
ρ14444-211-2-210041-2-2-211000000    orthogonal lifted from C32⋊C4
ρ154-44-211-2-2100-4-1222-1-1000000    symplectic lifted from C322C8, Schur index 2
ρ164-441-2-211-200-42-1-1-122000000    symplectic lifted from C322C8, Schur index 2
ρ1744-211-2-1+3-3/2-1-3-3/2100-211-1+3-3/2-1-3-3/21-2000000    complex lifted from C33⋊C4
ρ1844-211-2-1-3-3/2-1+3-3/2100-211-1-3-3/2-1+3-3/21-2000000    complex lifted from C33⋊C4
ρ194-4-2-2-1+3-3/2111-1-3-3/20021-3-3/22-1-11+3-3/2-1000000    complex faithful
ρ2044-2-2-1-3-3/2111-1+3-3/200-2-1-3-3/2-211-1+3-3/21000000    complex lifted from C33⋊C4
ρ214-4-2-2-1-3-3/2111-1+3-3/20021+3-3/22-1-11-3-3/2-1000000    complex faithful
ρ224-4-211-2-1+3-3/2-1-3-3/21002-1-11-3-3/21+3-3/2-12000000    complex faithful
ρ234-4-211-2-1-3-3/2-1+3-3/21002-1-11+3-3/21-3-3/2-12000000    complex faithful
ρ2444-2-2-1+3-3/2111-1-3-3/200-2-1+3-3/2-211-1-3-3/21000000    complex lifted from C33⋊C4

Permutation representations of C334C8
On 24 points - transitive group 24T552
Generators in S24
(2 22 16)(4 10 24)(6 18 12)(8 14 20)
(1 15 21)(2 22 16)(3 23 9)(4 10 24)(5 11 17)(6 18 12)(7 19 13)(8 14 20)
(1 21 15)(2 16 22)(3 23 9)(4 10 24)(5 17 11)(6 12 18)(7 19 13)(8 14 20)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)

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

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

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

G:=TransitiveGroup(24,552);

C334C8 is a maximal subgroup of
S3×C322C8  C335(C2×C8)  C33⋊M4(2)  C332M4(2)  C33⋊D8  C336SD16  C337SD16  C33⋊Q16  C337(C2×C8)  C334M4(2)  C3312M4(2)
C334C8 is a maximal quotient of
C334C16

Matrix representation of C334C8 in GL4(𝔽7) generated by

3243
4556
3361
0001
,
0526
0202
3361
0004
,
3632
6342
0020
0004
,
5331
2216
2565
3341
G:=sub<GL(4,GF(7))| [3,4,3,0,2,5,3,0,4,5,6,0,3,6,1,1],[0,0,3,0,5,2,3,0,2,0,6,0,6,2,1,4],[3,6,0,0,6,3,0,0,3,4,2,0,2,2,0,4],[5,2,2,3,3,2,5,3,3,1,6,4,1,6,5,1] >;

C334C8 in GAP, Magma, Sage, TeX

C_3^3\rtimes_4C_8
% in TeX

G:=Group("C3^3:4C8");
// GroupNames label

G:=SmallGroup(216,118);
// by ID

G=gap.SmallGroup(216,118);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,3,-3,12,31,963,201,964,730,5189]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^3=d^8=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^-1,b*c=c*b,d*b*d^-1=a^-1*b^-1,d*c*d^-1=c^-1>;
// generators/relations

Export

Subgroup lattice of C334C8 in TeX
Character table of C334C8 in TeX

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