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## G = C2×F8order 112 = 24·7

### Direct product of C2 and F8

Aliases: C2×F8, C24⋊C7, C23⋊C14, SmallGroup(112,41)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23 — C2×F8
 Chief series C1 — C23 — F8 — C2×F8
 Lower central C23 — C2×F8
 Upper central C1 — C2

Generators and relations for C2×F8
G = < a,b,c,d,e | a2=b2=c2=d2=e7=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=dc=cd, ece-1=b, ede-1=c >

Character table of C2×F8

 class 1 2A 2B 2C 7A 7B 7C 7D 7E 7F 14A 14B 14C 14D 14E 14F size 1 1 7 7 8 8 8 8 8 8 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ3 1 -1 -1 1 ζ72 ζ7 ζ73 ζ75 ζ74 ζ76 -ζ72 -ζ74 -ζ76 -ζ7 -ζ73 -ζ75 linear of order 14 ρ4 1 1 1 1 ζ75 ζ76 ζ74 ζ72 ζ73 ζ7 ζ75 ζ73 ζ7 ζ76 ζ74 ζ72 linear of order 7 ρ5 1 1 1 1 ζ74 ζ72 ζ76 ζ73 ζ7 ζ75 ζ74 ζ7 ζ75 ζ72 ζ76 ζ73 linear of order 7 ρ6 1 -1 -1 1 ζ76 ζ73 ζ72 ζ7 ζ75 ζ74 -ζ76 -ζ75 -ζ74 -ζ73 -ζ72 -ζ7 linear of order 14 ρ7 1 1 1 1 ζ7 ζ74 ζ75 ζ76 ζ72 ζ73 ζ7 ζ72 ζ73 ζ74 ζ75 ζ76 linear of order 7 ρ8 1 -1 -1 1 ζ74 ζ72 ζ76 ζ73 ζ7 ζ75 -ζ74 -ζ7 -ζ75 -ζ72 -ζ76 -ζ73 linear of order 14 ρ9 1 -1 -1 1 ζ73 ζ75 ζ7 ζ74 ζ76 ζ72 -ζ73 -ζ76 -ζ72 -ζ75 -ζ7 -ζ74 linear of order 14 ρ10 1 1 1 1 ζ72 ζ7 ζ73 ζ75 ζ74 ζ76 ζ72 ζ74 ζ76 ζ7 ζ73 ζ75 linear of order 7 ρ11 1 -1 -1 1 ζ7 ζ74 ζ75 ζ76 ζ72 ζ73 -ζ7 -ζ72 -ζ73 -ζ74 -ζ75 -ζ76 linear of order 14 ρ12 1 1 1 1 ζ73 ζ75 ζ7 ζ74 ζ76 ζ72 ζ73 ζ76 ζ72 ζ75 ζ7 ζ74 linear of order 7 ρ13 1 -1 -1 1 ζ75 ζ76 ζ74 ζ72 ζ73 ζ7 -ζ75 -ζ73 -ζ7 -ζ76 -ζ74 -ζ72 linear of order 14 ρ14 1 1 1 1 ζ76 ζ73 ζ72 ζ7 ζ75 ζ74 ζ76 ζ75 ζ74 ζ73 ζ72 ζ7 linear of order 7 ρ15 7 -7 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful ρ16 7 7 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from F8

Permutation representations of C2×F8
On 14 points - transitive group 14T9
Generators in S14
(1 12)(2 13)(3 14)(4 8)(5 9)(6 10)(7 11)
(1 12)(4 8)(6 10)(7 11)
(1 12)(2 13)(5 9)(7 11)
(1 12)(2 13)(3 14)(6 10)
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)

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

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

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

G:=TransitiveGroup(14,9);

On 16 points - transitive group 16T196
Generators in S16
(1 2)(3 13)(4 14)(5 15)(6 16)(7 10)(8 11)(9 12)
(1 4)(2 14)(3 8)(5 9)(6 7)(10 16)(11 13)(12 15)
(1 5)(2 15)(3 6)(4 9)(7 8)(10 11)(12 14)(13 16)
(1 6)(2 16)(3 5)(4 7)(8 9)(10 14)(11 12)(13 15)
(3 4 5 6 7 8 9)(10 11 12 13 14 15 16)

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

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

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

G:=TransitiveGroup(16,196);

On 28 points - transitive group 28T19
Generators in S28
(1 24)(2 25)(3 26)(4 27)(5 28)(6 22)(7 23)(8 17)(9 18)(10 19)(11 20)(12 21)(13 15)(14 16)
(1 17)(2 9)(4 27)(5 12)(6 22)(7 16)(8 24)(11 20)(13 15)(14 23)(18 25)(21 28)
(1 17)(2 18)(3 10)(5 28)(6 13)(7 23)(8 24)(9 25)(12 21)(14 16)(15 22)(19 26)
(1 24)(2 18)(3 19)(4 11)(6 22)(7 14)(8 17)(9 25)(10 26)(13 15)(16 23)(20 27)
(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)

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

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

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

G:=TransitiveGroup(28,19);

On 28 points - transitive group 28T20
Generators in S28
(1 24)(2 25)(3 26)(4 27)(5 28)(6 22)(7 23)(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)
(1 19)(4 15)(6 17)(7 18)(8 27)(10 22)(11 23)(12 24)
(1 19)(2 20)(5 16)(7 18)(9 28)(11 23)(12 24)(13 25)
(1 19)(2 20)(3 21)(6 17)(10 22)(12 24)(13 25)(14 26)
(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)

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

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

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

G:=TransitiveGroup(28,20);

Polynomial with Galois group C2×F8 over ℚ
actionf(x)Disc(f)
14T9x14+7x12-49x10-245x8+588x6+294x4-7214·725·194·318·5094

Matrix representation of C2×F8 in GL7(ℤ)

 -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 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

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],[0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0] >;

C2×F8 in GAP, Magma, Sage, TeX

C_2\times F_8
% in TeX

G:=Group("C2xF8");
// GroupNames label

G:=SmallGroup(112,41);
// by ID

G=gap.SmallGroup(112,41);
# by ID

G:=PCGroup([5,-2,-7,-2,2,2,217,568,884]);
// Polycyclic

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

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