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G = C339(C2×C4)  order 216 = 23·33

6th semidirect product of C33 and C2×C4 acting via C2×C4/C2=C22

metabelian, supersoluble, monomial, A-group

Aliases: C6.22S32, C339(C2×C4), C3⋊S32Dic3, C3⋊Dic35S3, C328(C4×S3), C32(S3×Dic3), (C3×C6).35D6, C32(C6.D6), C326(C2×Dic3), C2.1(C324D6), (C32×C6).13C22, (C3×C3⋊S3)⋊4C4, (C2×C3⋊S3).2S3, (C6×C3⋊S3).3C2, (C3×C3⋊Dic3)⋊6C2, SmallGroup(216,131)

Series: Derived Chief Lower central Upper central

C1C33 — C339(C2×C4)
C1C3C32C33C32×C6C6×C3⋊S3 — C339(C2×C4)
C33 — C339(C2×C4)
C1C2

Generators and relations for C339(C2×C4)
 G = < a,b,c,d,e | a3=b3=c3=d2=e4=1, ab=ba, ac=ca, dad=eae-1=a-1, bc=cb, bd=db, ebe-1=b-1, dcd=c-1, ce=ec, de=ed >

Subgroups: 340 in 90 conjugacy classes, 27 normal (11 characteristic)
C1, C2, C2 [×2], C3, C3 [×2], C3 [×4], C4 [×2], C22, S3 [×6], C6, C6 [×2], C6 [×6], C2×C4, C32, C32 [×2], C32 [×4], Dic3 [×6], C12 [×2], D6 [×3], C2×C6, C3×S3 [×6], C3⋊S3 [×2], C3×C6, C3×C6 [×2], C3×C6 [×4], C4×S3 [×2], C2×Dic3, C33, C3×Dic3 [×6], C3⋊Dic3 [×2], S3×C6 [×3], C2×C3⋊S3, C3×C3⋊S3 [×2], C32×C6, S3×Dic3 [×2], C6.D6, C3×C3⋊Dic3 [×2], C6×C3⋊S3, C339(C2×C4)
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×3], C2×C4, Dic3 [×2], D6 [×3], C4×S3 [×2], C2×Dic3, S32 [×3], S3×Dic3 [×2], C6.D6, C324D6, C339(C2×C4)

Character table of C339(C2×C4)

 class 12A2B2C3A3B3C3D3E3F3G3H4A4B4C4D6A6B6C6D6E6F6G6H6I6J12A12B12C12D
 size 119922244444999922244444181818181818
ρ1111111111111111111111111111111    trivial
ρ2111111111111-1-1-1-11111111111-1-1-1-1    linear of order 2
ρ311-1-111111111-111-111111111-1-11-11-1    linear of order 2
ρ411-1-1111111111-1-1111111111-1-1-11-11    linear of order 2
ρ51-11-111111111ii-i-i-1-1-1-1-1-1-1-1-11-i-iii    linear of order 4
ρ61-11-111111111-i-iii-1-1-1-1-1-1-1-1-11ii-i-i    linear of order 4
ρ71-1-1111111111-ii-ii-1-1-1-1-1-1-1-11-1-iii-i    linear of order 4
ρ81-1-1111111111i-ii-i-1-1-1-1-1-1-1-11-1i-i-ii    linear of order 4
ρ922-2-222-1-1-12-1-100002-12-1-1-12-1110000    orthogonal lifted from D6
ρ102200-1222-1-1-1-1-200-222-1-1-1-1-12000101    orthogonal lifted from D6
ρ1122002-12-1-1-12-10-2-20-1222-1-1-1-1001010    orthogonal lifted from D6
ρ12222222-1-1-12-1-100002-12-1-1-12-1-1-10000    orthogonal lifted from S3
ρ132200-1222-1-1-1-1200222-1-1-1-1-12000-10-1    orthogonal lifted from S3
ρ1422002-12-1-1-12-10220-1222-1-1-1-100-10-10    orthogonal lifted from S3
ρ152-22-222-1-1-12-1-10000-21-2111-211-10000    symplectic lifted from Dic3, Schur index 2
ρ162-2-2222-1-1-12-1-10000-21-2111-21-110000    symplectic lifted from Dic3, Schur index 2
ρ172-200-1222-1-1-1-12i00-2i-2-211111-2000i0-i    complex lifted from C4×S3
ρ182-2002-12-1-1-12-102i-2i01-2-2-2111100i0-i0    complex lifted from C4×S3
ρ192-200-1222-1-1-1-1-2i002i-2-211111-2000-i0i    complex lifted from C4×S3
ρ202-2002-12-1-1-12-10-2i2i01-2-2-2111100-i0i0    complex lifted from C4×S3
ρ2144004-2-211-2-210000-2-24-211-21000000    orthogonal lifted from S32
ρ224400-2-24-211-210000-24-2-2111-2000000    orthogonal lifted from S32
ρ234-400-2-24-211-2100002-422-1-1-12000000    orthogonal lifted from C6.D6
ρ244400-24-2-21-21100004-2-2111-2-2000000    orthogonal lifted from S32
ρ254-400-24-2-21-2110000-422-1-1-122000000    symplectic lifted from S3×Dic3, Schur index 2
ρ264-4004-2-211-2-21000022-42-1-12-1000000    symplectic lifted from S3×Dic3, Schur index 2
ρ274400-2-2-21-1-3-3/211-1+3-3/20000-2-2-21-1+3-3/2-1-3-3/211000000    complex lifted from C324D6
ρ284-400-2-2-21-1-3-3/211-1+3-3/20000222-11-3-3/21+3-3/2-1-1000000    complex faithful
ρ294-400-2-2-21-1+3-3/211-1-3-3/20000222-11+3-3/21-3-3/2-1-1000000    complex faithful
ρ304400-2-2-21-1+3-3/211-1-3-3/20000-2-2-21-1-3-3/2-1+3-3/211000000    complex lifted from C324D6

Permutation representations of C339(C2×C4)
On 24 points - transitive group 24T549
Generators in S24
(1 16 17)(2 18 13)(3 14 19)(4 20 15)(5 10 23)(6 24 11)(7 12 21)(8 22 9)
(1 16 17)(2 18 13)(3 14 19)(4 20 15)(5 23 10)(6 11 24)(7 21 12)(8 9 22)
(1 17 16)(2 18 13)(3 19 14)(4 20 15)(5 10 23)(6 11 24)(7 12 21)(8 9 22)
(1 22)(2 23)(3 24)(4 21)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 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)| (1,16,17)(2,18,13)(3,14,19)(4,20,15)(5,10,23)(6,24,11)(7,12,21)(8,22,9), (1,16,17)(2,18,13)(3,14,19)(4,20,15)(5,23,10)(6,11,24)(7,21,12)(8,9,22), (1,17,16)(2,18,13)(3,19,14)(4,20,15)(5,10,23)(6,11,24)(7,12,21)(8,9,22), (1,22)(2,23)(3,24)(4,21)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(11,19)(12,20), (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( (1,16,17)(2,18,13)(3,14,19)(4,20,15)(5,10,23)(6,24,11)(7,12,21)(8,22,9), (1,16,17)(2,18,13)(3,14,19)(4,20,15)(5,23,10)(6,11,24)(7,21,12)(8,9,22), (1,17,16)(2,18,13)(3,19,14)(4,20,15)(5,10,23)(6,11,24)(7,12,21)(8,9,22), (1,22)(2,23)(3,24)(4,21)(5,13)(6,14)(7,15)(8,16)(9,17)(10,18)(11,19)(12,20), (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([(1,16,17),(2,18,13),(3,14,19),(4,20,15),(5,10,23),(6,24,11),(7,12,21),(8,22,9)], [(1,16,17),(2,18,13),(3,14,19),(4,20,15),(5,23,10),(6,11,24),(7,21,12),(8,9,22)], [(1,17,16),(2,18,13),(3,19,14),(4,20,15),(5,10,23),(6,11,24),(7,12,21),(8,9,22)], [(1,22),(2,23),(3,24),(4,21),(5,13),(6,14),(7,15),(8,16),(9,17),(10,18),(11,19),(12,20)], [(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(24,549);

C339(C2×C4) is a maximal subgroup of
S32⋊Dic3  C33⋊C4⋊C4  (C3×C6).8D12  (C3×C6).9D12  S32×Dic3  S3×C6.D6  D64S32  C335(C2×Q8)  D6.4S32  D6.3S32  C3⋊S34Dic6  C12⋊S312S3  C4×C324D6  C62.96D6  C6224D6
C339(C2×C4) is a maximal quotient of
C12.93S32  C3310M4(2)  C336C42  C62.84D6  C62.85D6

Matrix representation of C339(C2×C4) in GL4(𝔽7) generated by

5323
1330
4406
0004
,
3145
1335
0040
0002
,
6251
0405
4406
0002
,
0526
2021
3301
1630
,
1351
6266
1633
3341
G:=sub<GL(4,GF(7))| [5,1,4,0,3,3,4,0,2,3,0,0,3,0,6,4],[3,1,0,0,1,3,0,0,4,3,4,0,5,5,0,2],[6,0,4,0,2,4,4,0,5,0,0,0,1,5,6,2],[0,2,3,1,5,0,3,6,2,2,0,3,6,1,1,0],[1,6,1,3,3,2,6,3,5,6,3,4,1,6,3,1] >;

C339(C2×C4) in GAP, Magma, Sage, TeX

C_3^3\rtimes_9(C_2\times C_4)
% in TeX

G:=Group("C3^3:9(C2xC4)");
// GroupNames label

G:=SmallGroup(216,131);
// by ID

G=gap.SmallGroup(216,131);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,31,387,201,730,5189]);
// Polycyclic

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

Export

Character table of C339(C2×C4) in TeX

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