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G = C74F7order 294 = 2·3·72

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

metabelian, supersoluble, monomial, A-group

Aliases: C74F7, C725C6, (C7×D7)⋊4C3, D72(C7⋊C3), C72⋊C31C2, C72(C2×C7⋊C3), SmallGroup(294,12)

Series: Derived Chief Lower central Upper central

C1C72 — C74F7
C1C7C72C72⋊C3 — C74F7
C72 — C74F7
C1

Generators and relations for C74F7
 G = < a,b,c | a7=b7=c6=1, ab=ba, cac-1=a2, cbc-1=b5 >

7C2
49C3
2C7
2C7
2C7
49C6
7C14
7C7⋊C3
7C7⋊C3
14C7⋊C3
14C7⋊C3
14C7⋊C3
7F7
7C2×C7⋊C3

Character table of C74F7

 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 C74F7
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 C74F7 over ℚ
actionf(x)Disc(f)
14T14x14-31958x7+656356768-244·328·721·2958

Matrix representation of C74F7 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] >;

C74F7 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 C74F7 in TeX
Character table of C74F7 in TeX

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