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G = 2+ 1+4.3C6order 192 = 26·3

The non-split extension by 2+ 1+4 of C6 acting via C6/C2=C3

non-abelian, soluble, monomial

Aliases: 2+ 1+4.3C6, C4○D42A4, (C22×C4)⋊3A4, Q8.2(C2×A4), C23⋊A44C2, C23.7(C2×A4), C4.2(C22⋊A4), C2.C252C3, C2.5(C2×C22⋊A4), SmallGroup(192,1509)

Series: Derived Chief Lower central Upper central

C1C22+ 1+4 — 2+ 1+4.3C6
C1C2C232+ 1+4C23⋊A4 — 2+ 1+4.3C6
2+ 1+4 — 2+ 1+4.3C6
C1C4

Generators and relations for 2+ 1+4.3C6
 G = < a,b,c,d,e | a4=b2=d2=1, c2=e6=a2, bab=a-1, ac=ca, ad=da, eae-1=a2bc, bc=cb, bd=db, ebe-1=a2bcd, dcd=a2c, ece-1=a-1d, ede-1=a-1bd >

Subgroups: 567 in 165 conjugacy classes, 17 normal (7 characteristic)
C1, C2, C2 [×5], C3, C4, C4 [×5], C22 [×12], C6, C2×C4 [×20], D4 [×20], Q8 [×2], Q8 [×6], C23 [×3], C23 [×4], C12, A4 [×3], C22×C4 [×3], C22×C4 [×4], C2×D4 [×15], C2×Q8 [×5], C4○D4 [×2], C4○D4 [×26], SL2(𝔽3) [×2], C2×A4 [×3], C2×C4○D4 [×5], 2+ 1+4, 2+ 1+4 [×3], 2- 1+4 [×2], C4×A4 [×3], C4.A4 [×2], C2.C25, C23⋊A4, 2+ 1+4.3C6
Quotients: C1, C2, C3, C6, A4 [×5], C2×A4 [×5], C22⋊A4, C2×C22⋊A4, 2+ 1+4.3C6

Character table of 2+ 1+4.3C6

 class 12A2B2C2D2E2F3A3B4A4B4C4D4E4F4G6A6B12A12B12C12D
 size 116666616161166666161616161616
ρ11111111111111111111111    trivial
ρ211-1111-111-1-11-1-1-1111-1-1-1-1    linear of order 2
ρ31111111ζ3ζ321111111ζ3ζ32ζ32ζ3ζ32ζ3    linear of order 3
ρ411-1111-1ζ3ζ32-1-11-1-1-11ζ3ζ32ζ6ζ65ζ6ζ65    linear of order 6
ρ51111111ζ32ζ31111111ζ32ζ3ζ3ζ32ζ3ζ32    linear of order 3
ρ611-1111-1ζ32ζ3-1-11-1-1-11ζ32ζ3ζ65ζ6ζ65ζ6    linear of order 6
ρ733-1-1-1-1300333-1-1-1-1000000    orthogonal lifted from A4
ρ8333-1-1-1-10033-1-1-1-13000000    orthogonal lifted from A4
ρ9331-1-1-1-300-3-33111-1000000    orthogonal lifted from C2×A4
ρ1033-3-1-1-1100-3-3-11113000000    orthogonal lifted from C2×A4
ρ1133-13-1-1-10033-1-13-1-1000000    orthogonal lifted from A4
ρ12331-13-1100-3-3-111-3-1000000    orthogonal lifted from C2×A4
ρ13331-1-13100-3-3-1-311-1000000    orthogonal lifted from C2×A4
ρ143313-1-1100-3-3-11-31-1000000    orthogonal lifted from C2×A4
ρ1533-1-13-1-10033-1-1-13-1000000    orthogonal lifted from A4
ρ1633-1-1-13-10033-13-1-1-1000000    orthogonal lifted from A4
ρ174-40000011-4i4i00000-1-1ii-i-i    complex faithful
ρ184-400000114i-4i00000-1-1-i-iii    complex faithful
ρ194-400000ζ32ζ3-4i4i00000ζ6ζ65ζ4ζ3ζ4ζ32ζ43ζ3ζ43ζ32    complex faithful
ρ204-400000ζ3ζ32-4i4i00000ζ65ζ6ζ4ζ32ζ4ζ3ζ43ζ32ζ43ζ3    complex faithful
ρ214-400000ζ3ζ324i-4i00000ζ65ζ6ζ43ζ32ζ43ζ3ζ4ζ32ζ4ζ3    complex faithful
ρ224-400000ζ32ζ34i-4i00000ζ6ζ65ζ43ζ3ζ43ζ32ζ4ζ3ζ4ζ32    complex faithful

Permutation representations of 2+ 1+4.3C6
On 16 points - transitive group 16T423
Generators in S16
(1 7 3 13)(2 16 4 10)(5 15 11 9)(6 14 12 8)
(1 13)(2 10)(3 7)(4 16)(5 9)(6 14)(8 12)(11 15)
(1 5 3 11)(2 14 4 8)(6 16 12 10)(7 15 13 9)
(5 11)(6 12)(8 14)(9 15)
(1 2 3 4)(5 6 7 8 9 10 11 12 13 14 15 16)

G:=sub<Sym(16)| (1,7,3,13)(2,16,4,10)(5,15,11,9)(6,14,12,8), (1,13)(2,10)(3,7)(4,16)(5,9)(6,14)(8,12)(11,15), (1,5,3,11)(2,14,4,8)(6,16,12,10)(7,15,13,9), (5,11)(6,12)(8,14)(9,15), (1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16)>;

G:=Group( (1,7,3,13)(2,16,4,10)(5,15,11,9)(6,14,12,8), (1,13)(2,10)(3,7)(4,16)(5,9)(6,14)(8,12)(11,15), (1,5,3,11)(2,14,4,8)(6,16,12,10)(7,15,13,9), (5,11)(6,12)(8,14)(9,15), (1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16) );

G=PermutationGroup([(1,7,3,13),(2,16,4,10),(5,15,11,9),(6,14,12,8)], [(1,13),(2,10),(3,7),(4,16),(5,9),(6,14),(8,12),(11,15)], [(1,5,3,11),(2,14,4,8),(6,16,12,10),(7,15,13,9)], [(5,11),(6,12),(8,14),(9,15)], [(1,2,3,4),(5,6,7,8,9,10,11,12,13,14,15,16)])

G:=TransitiveGroup(16,423);

Matrix representation of 2+ 1+4.3C6 in GL4(𝔽5) generated by

0300
3000
0002
0020
,
1000
0400
0010
0004
,
0030
0002
3000
0200
,
0030
0002
2000
0300
,
2000
0030
0001
0100
G:=sub<GL(4,GF(5))| [0,3,0,0,3,0,0,0,0,0,0,2,0,0,2,0],[1,0,0,0,0,4,0,0,0,0,1,0,0,0,0,4],[0,0,3,0,0,0,0,2,3,0,0,0,0,2,0,0],[0,0,2,0,0,0,0,3,3,0,0,0,0,2,0,0],[2,0,0,0,0,0,0,1,0,3,0,0,0,0,1,0] >;

2+ 1+4.3C6 in GAP, Magma, Sage, TeX

2_+^{1+4}._3C_6
% in TeX

G:=Group("ES+(2,2).3C6");
// GroupNames label

G:=SmallGroup(192,1509);
// by ID

G=gap.SmallGroup(192,1509);
# by ID

G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,672,135,262,851,375,1524,1027]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^2=d^2=1,c^2=e^6=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e^-1=a^2*b*c,b*c=c*b,b*d=d*b,e*b*e^-1=a^2*b*c*d,d*c*d=a^2*c,e*c*e^-1=a^-1*d,e*d*e^-1=a^-1*b*d>;
// generators/relations

Export

Character table of 2+ 1+4.3C6 in TeX

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