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G = C2×AΓL1(𝔽8)  order 336 = 24·3·7

Direct product of C2 and AΓL1(𝔽8)

direct product, non-abelian, soluble, monomial, A-group

Aliases: C2×AΓL1(𝔽8), F8⋊C6, (C2×F8)⋊C3, C24⋊(C7⋊C3), C23⋊(C2×C7⋊C3), SmallGroup(336,210)

Series: Derived Chief Lower central Upper central

C1C23F8 — C2×AΓL1(𝔽8)
C1C23F8AΓL1(𝔽8) — C2×AΓL1(𝔽8)
F8 — C2×AΓL1(𝔽8)
C1C2

Generators and relations for C2×AΓL1(𝔽8)
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e7=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, ebe-1=dc=cd, ece-1=fcf-1=b, ede-1=c, fdf-1=bcd, fef-1=e4 >

7C2
7C2
28C3
8C7
7C22
7C22
21C22
28C6
28C6
28C6
8C14
8C7⋊C3
7C23
7C23
7A4
28C2×C6
8C2×C7⋊C3
7C2×A4
7C2×A4
7C2×A4
7C22×A4

Character table of C2×AΓL1(𝔽8)

 class 12A2B2C3A3B6A6B6C6D6E6F7A7B14A14B
 size 1177282828282828282824242424
ρ11111111111111111    trivial
ρ21-1-1111-111-1-1-111-1-1    linear of order 2
ρ31111ζ32ζ3ζ32ζ3ζ32ζ3ζ3ζ321111    linear of order 3
ρ41111ζ3ζ32ζ3ζ32ζ3ζ32ζ32ζ31111    linear of order 3
ρ51-1-11ζ32ζ3ζ6ζ3ζ32ζ65ζ65ζ611-1-1    linear of order 6
ρ61-1-11ζ3ζ32ζ65ζ32ζ3ζ6ζ6ζ6511-1-1    linear of order 6
ρ7333300000000-1--7/2-1+-7/2-1--7/2-1+-7/2    complex lifted from C7⋊C3
ρ83-3-3300000000-1--7/2-1+-7/21+-7/21--7/2    complex lifted from C2×C7⋊C3
ρ93-3-3300000000-1+-7/2-1--7/21--7/21+-7/2    complex lifted from C2×C7⋊C3
ρ10333300000000-1+-7/2-1--7/2-1+-7/2-1--7/2    complex lifted from C7⋊C3
ρ1177-1-111-1-1-11-110000    orthogonal lifted from AΓL1(𝔽8)
ρ127-71-1111-1-1-11-10000    orthogonal faithful
ρ1377-1-1ζ32ζ3ζ6ζ65ζ6ζ3ζ65ζ320000    complex lifted from AΓL1(𝔽8)
ρ147-71-1ζ3ζ32ζ3ζ6ζ65ζ6ζ32ζ650000    complex faithful
ρ157-71-1ζ32ζ3ζ32ζ65ζ6ζ65ζ3ζ60000    complex faithful
ρ1677-1-1ζ3ζ32ζ65ζ6ζ65ζ32ζ6ζ30000    complex lifted from AΓL1(𝔽8)

Permutation representations of C2×AΓL1(𝔽8)
On 14 points - transitive group 14T18
Generators in S14
(1 14)(2 8)(3 9)(4 10)(5 11)(6 12)(7 13)
(1 14)(2 8)(5 11)(7 13)
(1 14)(2 8)(3 9)(6 12)
(2 8)(3 9)(4 10)(7 13)
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)
(2 3 5)(4 7 6)(8 9 11)(10 13 12)

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

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

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

G:=TransitiveGroup(14,18);

On 16 points - transitive group 16T712
Generators in S16
(1 2)(3 14)(4 15)(5 16)(6 10)(7 11)(8 12)(9 13)
(1 6)(2 10)(3 5)(4 7)(8 9)(11 15)(12 13)(14 16)
(1 7)(2 11)(3 9)(4 6)(5 8)(10 15)(12 16)(13 14)
(1 8)(2 12)(3 4)(5 7)(6 9)(10 13)(11 16)(14 15)
(3 4 5 6 7 8 9)(10 11 12 13 14 15 16)
(3 8 4)(6 7 9)(10 11 13)(12 15 14)

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

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

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

G:=TransitiveGroup(16,712);

On 28 points - transitive group 28T44
Generators in S28
(1 27)(2 28)(3 22)(4 23)(5 24)(6 25)(7 26)(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)
(1 18)(2 19)(5 15)(7 17)(8 24)(10 26)(11 27)(12 28)
(1 18)(2 19)(3 20)(6 16)(9 25)(11 27)(12 28)(13 22)
(2 19)(3 20)(4 21)(7 17)(10 26)(12 28)(13 22)(14 23)
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)(22 23 24 25 26 27 28)
(2 3 5)(4 7 6)(8 12 13)(9 14 10)(15 19 20)(16 21 17)(22 24 28)(23 26 25)

G:=sub<Sym(28)| (1,27)(2,28)(3,22)(4,23)(5,24)(6,25)(7,26)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21), (1,18)(2,19)(5,15)(7,17)(8,24)(10,26)(11,27)(12,28), (1,18)(2,19)(3,20)(6,16)(9,25)(11,27)(12,28)(13,22), (2,19)(3,20)(4,21)(7,17)(10,26)(12,28)(13,22)(14,23), (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28), (2,3,5)(4,7,6)(8,12,13)(9,14,10)(15,19,20)(16,21,17)(22,24,28)(23,26,25)>;

G:=Group( (1,27)(2,28)(3,22)(4,23)(5,24)(6,25)(7,26)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21), (1,18)(2,19)(5,15)(7,17)(8,24)(10,26)(11,27)(12,28), (1,18)(2,19)(3,20)(6,16)(9,25)(11,27)(12,28)(13,22), (2,19)(3,20)(4,21)(7,17)(10,26)(12,28)(13,22)(14,23), (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28), (2,3,5)(4,7,6)(8,12,13)(9,14,10)(15,19,20)(16,21,17)(22,24,28)(23,26,25) );

G=PermutationGroup([(1,27),(2,28),(3,22),(4,23),(5,24),(6,25),(7,26),(8,15),(9,16),(10,17),(11,18),(12,19),(13,20),(14,21)], [(1,18),(2,19),(5,15),(7,17),(8,24),(10,26),(11,27),(12,28)], [(1,18),(2,19),(3,20),(6,16),(9,25),(11,27),(12,28),(13,22)], [(2,19),(3,20),(4,21),(7,17),(10,26),(12,28),(13,22),(14,23)], [(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21),(22,23,24,25,26,27,28)], [(2,3,5),(4,7,6),(8,12,13),(9,14,10),(15,19,20),(16,21,17),(22,24,28),(23,26,25)])

G:=TransitiveGroup(28,44);

Polynomial with Galois group C2×AΓL1(𝔽8) over ℚ
actionf(x)Disc(f)
14T18x14+5x12-11x10-25x8+27x6+23x4-17x2+1-238·7312

Matrix representation of C2×AΓL1(𝔽8) in GL7(ℤ)

-1000000
0-100000
00-10000
000-1000
0000-100
00000-10
000000-1
,
-1000000
0100000
0010000
000-1000
0000-100
00000-10
0000001
,
-1000000
0-100000
0010000
000-1000
0000100
0000010
000000-1
,
1000000
0100000
00-10000
000-1000
0000-100
0000010
000000-1
,
0000100
0000010
0000001
1000000
0100000
0010000
0001000
,
1000000
0000100
0100000
0000010
0010000
0000001
0001000

G:=sub<GL(7,Integers())| [-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1],[0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0] >;

C2×AΓL1(𝔽8) in GAP, Magma, Sage, TeX

C_2\times {\rm AGammaL}_1({\mathbb F}_8)
% in TeX

G:=Group("C2xAGammaL(1,8)");
// GroupNames label

G:=SmallGroup(336,210);
// by ID

G=gap.SmallGroup(336,210);
# by ID

G:=PCGroup([6,-2,-3,-7,-2,2,2,116,2529,351,1900,856,767,1277]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^7=f^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,e*b*e^-1=d*c=c*d,e*c*e^-1=f*c*f^-1=b,e*d*e^-1=c,f*d*f^-1=b*c*d,f*e*f^-1=e^4>;
// generators/relations

Export

Subgroup lattice of C2×AΓL1(𝔽8) in TeX
Character table of C2×AΓL1(𝔽8) in TeX

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