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G = C7⋊F5order 140 = 22·5·7

The semidirect product of C7 and F5 acting via F5/D5=C2

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C7⋊F5, C5⋊Dic7, C351C4, D5.D7, (C7×D5).1C2, SmallGroup(140,6)

Series: Derived Chief Lower central Upper central

C1C35 — C7⋊F5
C1C7C35C7×D5 — C7⋊F5
C35 — C7⋊F5
C1

Generators and relations for C7⋊F5
 G = < a,b,c | a7=b5=c4=1, ab=ba, cac-1=a-1, cbc-1=b3 >

5C2
35C4
5C14
7F5
5Dic7

Character table of C7⋊F5

 class 124A4B57A7B7C14A14B14C35A35B35C35D35E35F
 size 1535354222101010444444
ρ111111111111111111    trivial
ρ211-1-11111111111111    linear of order 2
ρ31-1i-i1111-1-1-1111111    linear of order 4
ρ41-1-ii1111-1-1-1111111    linear of order 4
ρ522002ζ767ζ7473ζ7572ζ767ζ7572ζ7473ζ767ζ7473ζ7572ζ7572ζ7473ζ767    orthogonal lifted from D7
ρ622002ζ7572ζ767ζ7473ζ7572ζ7473ζ767ζ7572ζ767ζ7473ζ7473ζ767ζ7572    orthogonal lifted from D7
ρ722002ζ7473ζ7572ζ767ζ7473ζ767ζ7572ζ7473ζ7572ζ767ζ767ζ7572ζ7473    orthogonal lifted from D7
ρ82-2002ζ767ζ7473ζ757276775727473ζ767ζ7473ζ7572ζ7572ζ7473ζ767    symplectic lifted from Dic7, Schur index 2
ρ92-2002ζ7473ζ7572ζ76774737677572ζ7473ζ7572ζ767ζ767ζ7572ζ7473    symplectic lifted from Dic7, Schur index 2
ρ102-2002ζ7572ζ767ζ747375727473767ζ7572ζ767ζ7473ζ7473ζ767ζ7572    symplectic lifted from Dic7, Schur index 2
ρ114000-1444000-1-1-1-1-1-1    orthogonal lifted from F5
ρ124000-176+2ζ774+2ζ7375+2ζ72000ζ54ζ7654ζ75ζ765ζ7753ζ7453ζ7352ζ7452ζ737454ζ7554ζ725ζ755ζ7275ζ54ζ7554ζ725ζ755ζ7272ζ53ζ7453ζ7352ζ7452ζ7373ζ53ζ7653ζ752ζ7652ζ77    complex faithful
ρ134000-175+2ζ7276+2ζ774+2ζ7300054ζ7554ζ725ζ755ζ7275ζ54ζ7654ζ75ζ765ζ77ζ53ζ7453ζ7352ζ7452ζ737353ζ7453ζ7352ζ7452ζ7374ζ53ζ7653ζ752ζ7652ζ77ζ54ζ7554ζ725ζ755ζ7272    complex faithful
ρ144000-174+2ζ7375+2ζ7276+2ζ7000ζ53ζ7453ζ7352ζ7452ζ737354ζ7554ζ725ζ755ζ7275ζ53ζ7653ζ752ζ7652ζ77ζ54ζ7654ζ75ζ765ζ77ζ54ζ7554ζ725ζ755ζ727253ζ7453ζ7352ζ7452ζ7374    complex faithful
ρ154000-174+2ζ7375+2ζ7276+2ζ700053ζ7453ζ7352ζ7452ζ7374ζ54ζ7554ζ725ζ755ζ7272ζ54ζ7654ζ75ζ765ζ77ζ53ζ7653ζ752ζ7652ζ7754ζ7554ζ725ζ755ζ7275ζ53ζ7453ζ7352ζ7452ζ7373    complex faithful
ρ164000-176+2ζ774+2ζ7375+2ζ72000ζ53ζ7653ζ752ζ7652ζ77ζ53ζ7453ζ7352ζ7452ζ7373ζ54ζ7554ζ725ζ755ζ727254ζ7554ζ725ζ755ζ727553ζ7453ζ7352ζ7452ζ7374ζ54ζ7654ζ75ζ765ζ77    complex faithful
ρ174000-175+2ζ7276+2ζ774+2ζ73000ζ54ζ7554ζ725ζ755ζ7272ζ53ζ7653ζ752ζ7652ζ7753ζ7453ζ7352ζ7452ζ7374ζ53ζ7453ζ7352ζ7452ζ7373ζ54ζ7654ζ75ζ765ζ7754ζ7554ζ725ζ755ζ7275    complex faithful

Smallest permutation representation of C7⋊F5
On 35 points
Generators in S35
(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)(29 30 31 32 33 34 35)
(1 13 20 27 34)(2 14 21 28 35)(3 8 15 22 29)(4 9 16 23 30)(5 10 17 24 31)(6 11 18 25 32)(7 12 19 26 33)
(2 7)(3 6)(4 5)(8 18 29 25)(9 17 30 24)(10 16 31 23)(11 15 32 22)(12 21 33 28)(13 20 34 27)(14 19 35 26)

G:=sub<Sym(35)| (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)(29,30,31,32,33,34,35), (1,13,20,27,34)(2,14,21,28,35)(3,8,15,22,29)(4,9,16,23,30)(5,10,17,24,31)(6,11,18,25,32)(7,12,19,26,33), (2,7)(3,6)(4,5)(8,18,29,25)(9,17,30,24)(10,16,31,23)(11,15,32,22)(12,21,33,28)(13,20,34,27)(14,19,35,26)>;

G:=Group( (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)(29,30,31,32,33,34,35), (1,13,20,27,34)(2,14,21,28,35)(3,8,15,22,29)(4,9,16,23,30)(5,10,17,24,31)(6,11,18,25,32)(7,12,19,26,33), (2,7)(3,6)(4,5)(8,18,29,25)(9,17,30,24)(10,16,31,23)(11,15,32,22)(12,21,33,28)(13,20,34,27)(14,19,35,26) );

G=PermutationGroup([(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),(29,30,31,32,33,34,35)], [(1,13,20,27,34),(2,14,21,28,35),(3,8,15,22,29),(4,9,16,23,30),(5,10,17,24,31),(6,11,18,25,32),(7,12,19,26,33)], [(2,7),(3,6),(4,5),(8,18,29,25),(9,17,30,24),(10,16,31,23),(11,15,32,22),(12,21,33,28),(13,20,34,27),(14,19,35,26)])

C7⋊F5 is a maximal subgroup of   D7×F5  C35⋊C12  C5⋊Dic21
C7⋊F5 is a maximal quotient of   C35⋊C8  C5⋊Dic21

Matrix representation of C7⋊F5 in GL4(𝔽281) generated by

280100
3924100
002801
0039241
,
2672012800
252150280
2682012800
252160280
,
2794254187
5825422327
40100
8724100
G:=sub<GL(4,GF(281))| [280,39,0,0,1,241,0,0,0,0,280,39,0,0,1,241],[267,252,268,252,201,15,201,16,280,0,280,0,0,280,0,280],[27,58,40,87,94,254,1,241,254,223,0,0,187,27,0,0] >;

C7⋊F5 in GAP, Magma, Sage, TeX

C_7\rtimes F_5
% in TeX

G:=Group("C7:F5");
// GroupNames label

G:=SmallGroup(140,6);
// by ID

G=gap.SmallGroup(140,6);
# by ID

G:=PCGroup([4,-2,-2,-5,-7,8,146,102,1923]);
// Polycyclic

G:=Group<a,b,c|a^7=b^5=c^4=1,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^3>;
// generators/relations

Export

Subgroup lattice of C7⋊F5 in TeX
Character table of C7⋊F5 in TeX

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