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G = C7⋊4F7  order 294 = 2·3·72

2nd semidirect product of C7 and F7 acting via F7/D7=C3

metabelian, supersoluble, monomial, A-group

Aliases: C7⋊4F7, C72⋊5C6, (C7×D7)⋊4C3, D7⋊2(C7⋊C3), C72⋊C3⋊1C2, C7⋊2(C2×C7⋊C3), SmallGroup(294,12)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C72 — C7⋊4F7
Chief series
C1 — C7 — C72 — C72⋊C3 — C7⋊4F7
Lower central
C72 — C7⋊4F7
Upper central
C1

Generators and relations for C7⋊4F7
 G = < a,b,c | a7=b7=c6=1, ab=ba, cac-1=a2, cbc-1=b5 >

C1
7C2
49C3
C7
C7
2C7
2C7
2C7
49C6
D7
7C14
7C7⋊C3
7C7⋊C3
14C7⋊C3
14C7⋊C3
14C7⋊C3
C72
7F7
7C2×C7⋊C3
C7×D7
C72⋊C3
C7⋊4F7

Character table of C7⋊4F7

 class 123A3B6A6B7A7B7C7D7E7F7G7H7I14A14B
 size 17494949493366666662121
ρ111111111111111111    trivial
ρ21-111-1-1111111111-1-1    linear of order 2
ρ311ζ32ζ3ζ32ζ311111111111    linear of order 3
ρ411ζ3ζ32ζ3ζ3211111111111    linear of order 3
ρ51-1ζ32ζ3ζ6ζ65111111111-1-1    linear of order 6
ρ61-1ζ3ζ32ζ65ζ6111111111-1-1    linear of order 6
ρ7330000-1-√-7/2-1+√-7/2-1-√-7/2-1-√-7/23-1-√-7/2-1+√-7/2-1+√-7/2-1+√-7/2-1+√-7/2-1-√-7/2    complex lifted from C7⋊C3
ρ8330000-1+√-7/2-1-√-7/2-1+√-7/2-1+√-7/23-1+√-7/2-1-√-7/2-1-√-7/2-1-√-7/2-1-√-7/2-1+√-7/2    complex lifted from C7⋊C3
ρ93-30000-1-√-7/2-1+√-7/2-1-√-7/2-1-√-7/23-1-√-7/2-1+√-7/2-1+√-7/2-1+√-7/21-√-7/21+√-7/2    complex lifted from C2×C7⋊C3
ρ103-30000-1+√-7/2-1-√-7/2-1+√-7/2-1+√-7/23-1+√-7/2-1-√-7/2-1-√-7/2-1-√-7/21+√-7/21-√-7/2    complex lifted from C2×C7⋊C3
ρ1160000066-1-1-1-1-1-1-100    orthogonal lifted from F7
ρ12600000-1-√-7-1+√-7-1+√-7-1-15-√-7/2-15+√-7/2-1-√-700    complex faithful
ρ13600000-1+√-7-1-√-75+√-7/2-1-√-7-1-1-1+√-7-15-√-7/200    complex faithful
ρ14600000-1+√-7-1-√-7-1-√-7-1-15+√-7/2-15-√-7/2-1+√-700    complex faithful
ρ15600000-1+√-7-1-√-7-15+√-7/2-1-1-√-75-√-7/2-1+√-7-100    complex faithful
ρ16600000-1-√-7-1+√-7-15-√-7/2-1-1+√-75+√-7/2-1-√-7-100    complex faithful
ρ17600000-1-√-7-1+√-75-√-7/2-1+√-7-1-1-1-√-7-15+√-7/200    complex faithful

Permutation representations of C7⋊4F7
►On 14 points - transitive group 14T14
Generators in S14
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)

(1 5 2 6 3 7 4)(8 11 14 10 13 9 12)

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

Polynomial with Galois group C7⋊4F7 over ℚ
actionf(x)Disc(f)
14T14x14-31958x7+656356768-244·328·721·2958

Matrix representation of C7⋊4F7 ►in GL6(𝔽2)

110100
110011
101010
111100
110010
001011
,
101111
011110
110111
101010
011010
111100
,
100101
001100
000111
010000
010101
000001

G:=sub<GL(6,GF(2))| [1,1,1,1,1,0,1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,1,0,0,0,1,1,0,1,1,0,1,0,0,0,1],[1,0,1,1,0,1,0,1,1,0,1,1,1,1,0,1,1,1,1,1,1,0,0,1,1,1,1,1,1,0,1,0,1,0,0,0],[1,0,0,0,0,0,0,0,0,1,1,0,0,1,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,0,1,0,1,0,1,1] >;
 

C7⋊4F7 in GAP, Magma, Sage, TeX 

C_7\rtimes_4F_7
 
% in TeX
 
G:=Group("C7:4F7");
 
// GroupNames label
 
G:=SmallGroup(294,12);
 
// by ID
 
G=gap.SmallGroup(294,12);
 
# by ID
 
G:=PCGroup([4,-2,-3,-7,-7,150,4035,1351]);
 
// Polycyclic
 
G:=Group<a,b,c|a^7=b^7=c^6=1,a*b=b*a,c*a*c^-1=a^2,c*b*c^-1=b^5>;
 
// generators/relations
 

Export

Subgroup lattice of C7⋊4F7 in TeX
Character table of C7⋊4F7 in TeX

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