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G = 2+ 1+4.C6order 192 = 26·3

1st non-split extension by 2+ 1+4 of C6 acting faithfully

non-abelian, soluble, monomial

Aliases: 2+ 1+4.1C6, (C22×C4)⋊2A4, C23.7D4⋊C3, C23.2(C2×A4), C23⋊A4.1C2, C2.5(C24⋊C6), SmallGroup(192,202)

Series: Derived Chief Lower central Upper central

C1C22+ 1+4 — 2+ 1+4.C6
C1C2C232+ 1+4C23⋊A4 — 2+ 1+4.C6
2+ 1+4 — 2+ 1+4.C6
C1C2

Generators and relations for 2+ 1+4.C6
 G = < a,b,c,d,e | a4=b2=d2=1, c2=e6=a2, bab=ece-1=a-1, ac=ca, ad=da, eae-1=acd, bc=cb, bd=db, ebe-1=a-1c, dcd=a2c, ede-1=a-1bd >

6C2
12C2
16C3
3C22
4C4
4C22
6C4
6C22
8C22
12C22
12C4
16C6
2Q8
2C23
3C23
3C2×C4
6C2×C4
6D4
6C2×C4
6D4
6C2×C4
6D4
4A4
8A4
16C12
3C2×D4
3C22⋊C4
6C4⋊C4
6C4○D4
6C22⋊C4
6C2×D4
4C2×A4
8SL2(𝔽3)
8C2×A4
3C22.D4
3C23⋊C4
4C4×A4

Character table of 2+ 1+4.C6

 class 12A2B2C3A3B4A4B4C4D6A6B12A12B12C12D
 size 116121616441224161616161616
ρ11111111111111111    trivial
ρ2111111-1-11-111-1-1-1-1    linear of order 2
ρ31111ζ3ζ32-1-11-1ζ32ζ3ζ65ζ65ζ6ζ6    linear of order 6
ρ41111ζ3ζ321111ζ32ζ3ζ3ζ3ζ32ζ32    linear of order 3
ρ51111ζ32ζ31111ζ3ζ32ζ32ζ32ζ3ζ3    linear of order 3
ρ61111ζ32ζ3-1-11-1ζ3ζ32ζ6ζ6ζ65ζ65    linear of order 6
ρ7333-100-3-3-11000000    orthogonal lifted from C2×A4
ρ8333-10033-1-1000000    orthogonal lifted from A4
ρ94-400112i-2i00-1-1i-ii-i    complex faithful
ρ104-40011-2i2i00-1-1-ii-ii    complex faithful
ρ114-400ζ32ζ3-2i2i00ζ65ζ6ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3    complex faithful
ρ124-400ζ3ζ322i-2i00ζ6ζ65ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32    complex faithful
ρ134-400ζ3ζ32-2i2i00ζ6ζ65ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32    complex faithful
ρ144-400ζ32ζ32i-2i00ζ65ζ6ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3    complex faithful
ρ1566-2-2000020000000    orthogonal lifted from C24⋊C6
ρ1666-220000-20000000    orthogonal lifted from C24⋊C6

Permutation representations of 2+ 1+4.C6
On 16 points - transitive group 16T426
Generators in S16
(1 11 3 5)(2 14 4 8)(6 16 12 10)(7 15 13 9)
(1 5)(2 8)(3 11)(4 14)(6 10)(7 15)(9 13)(12 16)
(1 15 3 9)(2 6 4 12)(5 7 11 13)(8 10 14 16)
(1 3)(2 4)(5 11)(8 14)
(1 2 3 4)(5 6 7 8 9 10 11 12 13 14 15 16)

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

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

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

G:=TransitiveGroup(16,426);

On 16 points - transitive group 16T428
Generators in S16
(1 14 3 8)(2 9 4 15)(5 13 11 7)(6 16 12 10)
(2 13)(4 7)(5 9)(6 12)(8 14)(11 15)
(1 10 3 16)(2 9 4 15)(5 7 11 13)(6 8 12 14)
(1 16)(2 13)(3 10)(4 7)(5 15)(6 8)(9 11)(12 14)
(1 2 3 4)(5 6 7 8 9 10 11 12 13 14 15 16)

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

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

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

G:=TransitiveGroup(16,428);

Matrix representation of 2+ 1+4.C6 in GL4(𝔽5) generated by

0213
2130
1303
3034
,
1000
3420
0010
3034
,
0040
2021
1000
3420
,
4000
2021
0010
2130
,
3400
0040
3004
4000
G:=sub<GL(4,GF(5))| [0,2,1,3,2,1,3,0,1,3,0,3,3,0,3,4],[1,3,0,3,0,4,0,0,0,2,1,3,0,0,0,4],[0,2,1,3,0,0,0,4,4,2,0,2,0,1,0,0],[4,2,0,2,0,0,0,1,0,2,1,3,0,1,0,0],[3,0,3,4,4,0,0,0,0,4,0,0,0,0,4,0] >;

2+ 1+4.C6 in GAP, Magma, Sage, TeX

2_+^{1+4}.C_6
% in TeX

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

G:=SmallGroup(192,202);
// by ID

G=gap.SmallGroup(192,202);
# by ID

G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,672,632,135,1683,262,851,375,3540,1027]);
// Polycyclic

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

Export

Subgroup lattice of 2+ 1+4.C6 in TeX
Character table of 2+ 1+4.C6 in TeX

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