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G = C72⋊C6  order 294 = 2·3·72

7th semidirect product of C72 and C6 acting faithfully

metabelian, supersoluble, monomial, A-group

Aliases: C7⋊2F7, C72⋊7C6, C7⋊D7⋊3C3, C72⋊3C3⋊3C2, SmallGroup(294,14)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C72 — C72⋊C6
Chief series
C1 — C7 — C72 — C72⋊3C3 — C72⋊C6
Lower central
C72 — C72⋊C6
Upper central
C1

Generators and relations for C72⋊C6
 G = < a,b,c | a7=b7=c6=1, ab=ba, cac-1=a5, cbc-1=b3 >

C1
49C2
49C3
C7
C7
3C7
3C7
49C6
7D7
7D7
21D7
21D7
7C7⋊C3
7C7⋊C3
C72
7F7
7F7
C7⋊D7
C72⋊3C3
C72⋊C6

Character table of C72⋊C6

 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
ρ7600000-1-1-16-1-1-1-1    orthogonal lifted from F7
ρ8600000-1-1-1-16-1-1-1    orthogonal lifted from F7
ρ96000002ζ74+2ζ73+2ζ76+2ζ74+2ζ73+ζ72ζ76+ζ75+ζ72+2ζ7-1-12ζ75+2ζ72+22ζ76+2ζ7+22ζ75+ζ74+ζ73+2ζ72    orthogonal faithful
ρ106000002ζ75+ζ74+ζ73+2ζ722ζ76+2ζ7+22ζ75+2ζ72+2-1-12ζ76+ζ75+ζ72+2ζ7ζ76+2ζ74+2ζ73+ζ72ζ74+2ζ73+2    orthogonal faithful
ρ11600000ζ76+2ζ74+2ζ73+ζ72ζ75+2ζ72+22ζ74+2ζ73+2-1-12ζ75+ζ74+ζ73+2ζ722ζ76+ζ75+ζ72+2ζ72ζ76+2ζ7+2    orthogonal faithful
ρ126000002ζ76+2ζ7+22ζ76+ζ75+ζ72+2ζ72ζ75+ζ74+ζ73+2ζ72-1-12ζ74+2ζ73+22ζ75+2ζ72+2ζ76+2ζ74+2ζ73+ζ7    orthogonal faithful
ρ136000002ζ75+2ζ72+22ζ75+ζ74+ζ73+2ζ72ζ76+2ζ74+2ζ73+ζ7-1-12ζ76+2ζ7+22ζ74+2ζ73+22ζ76+ζ75+ζ72+2ζ7    orthogonal faithful
ρ146000002ζ76+ζ75+ζ72+2ζ72ζ74+2ζ73+22ζ76+2ζ7+2-1-1ζ76+2ζ74+2ζ73+ζ72ζ75+ζ74+ζ73+2ζ722ζ75+2ζ72+2    orthogonal faithful

Permutation representations of C72⋊C6
►On 21 points - transitive group 21T19
Generators in S21
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)

(1 7 6 5 4 3 2)(8 13 11 9 14 12 10)(15 18 21 17 20 16 19)

(1 11 16)(2 14 18 7 8 21)(3 10 20 6 12 19)(4 13 15 5 9 17)
 
G:=sub<Sym(21)| (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21), (1,7,6,5,4,3,2)(8,13,11,9,14,12,10)(15,18,21,17,20,16,19), (1,11,16)(2,14,18,7,8,21)(3,10,20,6,12,19)(4,13,15,5,9,17)>;
 
G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21), (1,7,6,5,4,3,2)(8,13,11,9,14,12,10)(15,18,21,17,20,16,19), (1,11,16)(2,14,18,7,8,21)(3,10,20,6,12,19)(4,13,15,5,9,17) );
 
G=PermutationGroup([[(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21)], [(1,7,6,5,4,3,2),(8,13,11,9,14,12,10),(15,18,21,17,20,16,19)], [(1,11,16),(2,14,18,7,8,21),(3,10,20,6,12,19),(4,13,15,5,9,17)]])
 
G:=TransitiveGroup(21,19);
 

Matrix representation of C72⋊C6 ►in GL6(𝔽43)

28340000
9340000
0015100
0042000
2440372415
32390327
,
1510000
4200000
00283400
0093400
074152415
1902229327
,
242444246
202032321925
34280000
3490000
11090023
702828042

G:=sub<GL(6,GF(43))| [28,9,0,0,24,3,34,34,0,0,4,2,0,0,15,42,0,39,0,0,1,0,37,0,0,0,0,0,24,3,0,0,0,0,15,27],[15,42,0,0,0,19,1,0,0,0,7,0,0,0,28,9,4,22,0,0,34,34,15,29,0,0,0,0,24,3,0,0,0,0,15,27],[24,20,34,34,11,7,24,20,28,9,0,0,4,32,0,0,9,28,4,32,0,0,0,28,24,19,0,0,0,0,6,25,0,0,23,42] >;
 

C72⋊C6 in GAP, Magma, Sage, TeX 

C_7^2\rtimes C_6
 
% in TeX
 
G:=Group("C7^2:C6");
 
// GroupNames label
 
G:=SmallGroup(294,14);
 
// by ID
 
G=gap.SmallGroup(294,14);
 
# by ID
 
G:=PCGroup([4,-2,-3,-7,-7,434,78,4035,1351]);
 
// 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^3>;
 
// generators/relations
 

Export

Subgroup lattice of C72⋊C6 in TeX
Character table of C72⋊C6 in TeX

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