C2xF8 - GroupNames

G = C₂×F₈ order 112 = 2⁴·7

Direct product of C₂ and F₈

direct product, metabelian, soluble, monomial, A-group

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 >

C₁

C₂

₇C₂

₈C₇

₇C₂²

₈C₁₄

C₂³

₇C₂³

C₂⁴

F₈

C₂×F₈

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

action	f(x)	Disc(f)
14T9	x¹⁴+7x¹²-49x¹⁰-245x⁸+588x⁶+294x⁴-7	2¹⁴·7²⁵·19⁴·31⁸·509⁴

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

Export

Subgroup lattice of C₂×F₈ in TeX
Character table of C₂×F₈ in TeX

G = C2×F8 order 112 = 24·7

Direct product of C2 and F8

G = C₂×F₈ order 112 = 2⁴·7

Direct product of C₂ and F₈