Copied to
clipboard

G = C2×F8order 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

C1C23 — C2×F8
C1C23F8 — C2×F8
C23 — C2×F8
C1C2

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 >

7C2
7C2
8C7
7C22
7C22
7C22
7C22
7C22
8C14
7C23
7C23

Character table of C2×F8

 class 12A2B2C7A7B7C7D7E7F14A14B14C14D14E14F
 size 1177888888888888
ρ11111111111111111    trivial
ρ21-1-11111111-1-1-1-1-1-1    linear of order 2
ρ31-1-11ζ72ζ7ζ73ζ75ζ74ζ7672747677375    linear of order 14
ρ41111ζ75ζ76ζ74ζ72ζ73ζ7ζ75ζ73ζ7ζ76ζ74ζ72    linear of order 7
ρ51111ζ74ζ72ζ76ζ73ζ7ζ75ζ74ζ7ζ75ζ72ζ76ζ73    linear of order 7
ρ61-1-11ζ76ζ73ζ72ζ7ζ75ζ7476757473727    linear of order 14
ρ71111ζ7ζ74ζ75ζ76ζ72ζ73ζ7ζ72ζ73ζ74ζ75ζ76    linear of order 7
ρ81-1-11ζ74ζ72ζ76ζ73ζ7ζ7574775727673    linear of order 14
ρ91-1-11ζ73ζ75ζ7ζ74ζ76ζ7273767275774    linear of order 14
ρ101111ζ72ζ7ζ73ζ75ζ74ζ76ζ72ζ74ζ76ζ7ζ73ζ75    linear of order 7
ρ111-1-11ζ7ζ74ζ75ζ76ζ72ζ7377273747576    linear of order 14
ρ121111ζ73ζ75ζ7ζ74ζ76ζ72ζ73ζ76ζ72ζ75ζ7ζ74    linear of order 7
ρ131-1-11ζ75ζ76ζ74ζ72ζ73ζ775737767472    linear of order 14
ρ141111ζ76ζ73ζ72ζ7ζ75ζ74ζ76ζ75ζ74ζ73ζ72ζ7    linear of order 7
ρ157-71-1000000000000    orthogonal faithful
ρ1677-1-1000000000000    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 12)(4 13)(5 14)(6 15)(7 16)(8 10)(9 11)
(1 5)(2 14)(3 6)(4 9)(7 8)(10 16)(11 13)(12 15)
(1 6)(2 15)(3 5)(4 7)(8 9)(10 11)(12 14)(13 16)
(1 7)(2 16)(3 9)(4 6)(5 8)(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)(3,12)(4,13)(5,14)(6,15)(7,16)(8,10)(9,11), (1,5)(2,14)(3,6)(4,9)(7,8)(10,16)(11,13)(12,15), (1,6)(2,15)(3,5)(4,7)(8,9)(10,11)(12,14)(13,16), (1,7)(2,16)(3,9)(4,6)(5,8)(10,14)(11,12)(13,15), (3,4,5,6,7,8,9)(10,11,12,13,14,15,16)>;

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

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

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

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

׿
×
𝔽