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G = C7⋊F7  order 294 = 2·3·72

1st semidirect product of C7 and F7 acting via F7/C7=C6

metabelian, supersoluble, monomial, A-group

Aliases: C7⋊1F7, C72⋊6C6, C7⋊D7⋊2C3, C72⋊C3⋊2C2, SmallGroup(294,13)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for C7⋊F7
 G = < a,b,c | a7=b7=c6=1, ab=ba, cac-1=a5, cbc-1=b5 >

C1
49C2
49C3
C7
C7
C7
C7
C7
C7
C7
C7
49C6
7D7
7D7
7D7
7D7
7D7
7D7
7D7
7D7
7C7⋊C3
7C7⋊C3
7C7⋊C3
7C7⋊C3
7C7⋊C3
7C7⋊C3
7C7⋊C3
7C7⋊C3
C72
7F7
7F7
7F7
7F7
7F7
7F7
7F7
7F7
C7⋊D7
C72⋊C3
C7⋊F7

Character table of C7⋊F7

 class 123A3B6A6B7A7B7C7D7E7F7G7H
 size 1494949494966666666
ρ111111111111111    trivial
ρ21-111-1-111111111    linear of order 2
ρ311ζ32ζ3ζ32ζ311111111    linear of order 3
ρ411ζ3ζ32ζ3ζ3211111111    linear of order 3
ρ51-1ζ3ζ32ζ65ζ611111111    linear of order 6
ρ61-1ζ32ζ3ζ6ζ6511111111    linear of order 6
ρ7600000-1-1-1-1-1-16-1    orthogonal lifted from F7
ρ8600000-1-16-1-1-1-1-1    orthogonal lifted from F7
ρ9600000-1-1-1-1-16-1-1    orthogonal lifted from F7
ρ10600000-1-1-16-1-1-1-1    orthogonal lifted from F7
ρ11600000-1-1-1-16-1-1-1    orthogonal lifted from F7
ρ12600000-16-1-1-1-1-1-1    orthogonal lifted from F7
ρ13600000-1-1-1-1-1-1-16    orthogonal lifted from F7
ρ146000006-1-1-1-1-1-1-1    orthogonal lifted from F7

Smallest permutation representation of C7⋊F7
►On 49 points
Generators in S49
(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)(36 37 38 39 40 41 42)(43 44 45 46 47 48 49)

(1 47 32 26 11 36 18)(2 48 33 27 12 37 19)(3 49 34 28 13 38 20)(4 43 35 22 14 39 21)(5 44 29 23 8 40 15)(6 45 30 24 9 41 16)(7 46 31 25 10 42 17)

(2 4 3 7 5 6)(8 41 48 22 34 17)(9 37 43 28 31 15)(10 40 45 27 35 20)(11 36 47 26 32 18)(12 39 49 25 29 16)(13 42 44 24 33 21)(14 38 46 23 30 19)
 
G:=sub<Sym(49)| (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)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49), (1,47,32,26,11,36,18)(2,48,33,27,12,37,19)(3,49,34,28,13,38,20)(4,43,35,22,14,39,21)(5,44,29,23,8,40,15)(6,45,30,24,9,41,16)(7,46,31,25,10,42,17), (2,4,3,7,5,6)(8,41,48,22,34,17)(9,37,43,28,31,15)(10,40,45,27,35,20)(11,36,47,26,32,18)(12,39,49,25,29,16)(13,42,44,24,33,21)(14,38,46,23,30,19)>;
 
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)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49), (1,47,32,26,11,36,18)(2,48,33,27,12,37,19)(3,49,34,28,13,38,20)(4,43,35,22,14,39,21)(5,44,29,23,8,40,15)(6,45,30,24,9,41,16)(7,46,31,25,10,42,17), (2,4,3,7,5,6)(8,41,48,22,34,17)(9,37,43,28,31,15)(10,40,45,27,35,20)(11,36,47,26,32,18)(12,39,49,25,29,16)(13,42,44,24,33,21)(14,38,46,23,30,19) );
 
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),(36,37,38,39,40,41,42),(43,44,45,46,47,48,49)], [(1,47,32,26,11,36,18),(2,48,33,27,12,37,19),(3,49,34,28,13,38,20),(4,43,35,22,14,39,21),(5,44,29,23,8,40,15),(6,45,30,24,9,41,16),(7,46,31,25,10,42,17)], [(2,4,3,7,5,6),(8,41,48,22,34,17),(9,37,43,28,31,15),(10,40,45,27,35,20),(11,36,47,26,32,18),(12,39,49,25,29,16),(13,42,44,24,33,21),(14,38,46,23,30,19)]])
 

Matrix representation of C7⋊F7 ►in GL12(ℤ)

170000000000
0-11000000000
-1-10100000000
160010000000
0-10001000000
0-2-1-1-1-1000000
000000-110000
000000-101000
000000-100100
000000-100010
000000-100001
000000-100000
,
100000000000
010000000000
001000000000
000100000000
000010000000
000001000000
00000000000-1
00000010000-1
00000001000-1
00000000100-1
00000000010-1
00000000001-1
,
100000000000
000001000000
-100100000000
110000000000
0-1-1-1-1-1000000
000010000000
0000000-101-10
00000000-1100
000000-1001-10
00000000-110-1
000000-100000
0000000-1010-1

G:=sub<GL(12,Integers())| [1,0,-1,1,0,0,0,0,0,0,0,0,7,-1,-1,6,-1,-2,0,0,0,0,0,0,0,1,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,-1,-1,-1,-1],[1,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,-1,0,0,0,0,0,0,0,0,1,1,1,1,0,1,0,0,0,0,0,0,-1,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,-1,0,-1] >;
 

C7⋊F7 in GAP, Magma, Sage, TeX 

C_7\rtimes F_7
 
% in TeX
 
G:=Group("C7:F7");
 
// GroupNames label
 
G:=SmallGroup(294,13);
 
// by ID
 
G=gap.SmallGroup(294,13);
 
# by ID
 
G:=PCGroup([4,-2,-3,-7,-7,434,150,4035,1351]);
 
// Polycyclic
 
G:=Group<a,b,c|a^7=b^7=c^6=1,a*b=b*a,c*a*c^-1=a^5,c*b*c^-1=b^5>;
 
// generators/relations
 

Export

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

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