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G = C72⋊C6order 294 = 2·3·72

7th semidirect product of C72 and C6 acting faithfully

metabelian, supersoluble, monomial, A-group

Aliases: C72F7, C727C6, C7⋊D73C3, C723C33C2, SmallGroup(294,14)

Series: Derived Chief Lower central Upper central

C1C72 — C72⋊C6
C1C7C72C723C3 — C72⋊C6
C72 — C72⋊C6
C1

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

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

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
ρ960000074+2ζ73+2ζ76+2ζ74+2ζ737767572+2ζ7-1-175+2ζ72+276+2ζ7+2757473+2ζ72    orthogonal faithful
ρ10600000757473+2ζ7276+2ζ7+275+2ζ72+2-1-1767572+2ζ7ζ76+2ζ74+2ζ73774+2ζ73+2    orthogonal faithful
ρ11600000ζ76+2ζ74+2ζ73775+2ζ72+274+2ζ73+2-1-1757473+2ζ72767572+2ζ776+2ζ7+2    orthogonal faithful
ρ1260000076+2ζ7+2767572+2ζ7757473+2ζ72-1-174+2ζ73+275+2ζ72+2ζ76+2ζ74+2ζ737    orthogonal faithful
ρ1360000075+2ζ72+2757473+2ζ72ζ76+2ζ74+2ζ737-1-176+2ζ7+274+2ζ73+2767572+2ζ7    orthogonal faithful
ρ14600000767572+2ζ774+2ζ73+276+2ζ7+2-1-1ζ76+2ζ74+2ζ737757473+2ζ7275+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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