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G = C3×F7order 126 = 2·32·7

Direct product of C3 and F7

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C3×F7, D7⋊C32, C212C6, C7⋊C3⋊C6, C7⋊(C3×C6), (C3×D7)⋊C3, (C3×C7⋊C3)⋊2C2, SmallGroup(126,7)

Series: Derived Chief Lower central Upper central

C1C7 — C3×F7
C1C7C21C3×C7⋊C3 — C3×F7
C7 — C3×F7
C1C3

Generators and relations for C3×F7
 G = < a,b,c | a3=b7=c6=1, ab=ba, ac=ca, cbc-1=b5 >

7C2
7C3
7C3
7C3
7C6
7C6
7C6
7C6
7C32
7C3×C6

Character table of C3×F7

 class 123A3B3C3D3E3F3G3H6A6B6C6D6E6F6G6H721A21B
 size 171177777777777777666
ρ1111111111111111111111    trivial
ρ21-111111111-1-1-1-1-1-1-1-1111    linear of order 2
ρ31-1ζ32ζ311ζ32ζ3ζ3ζ32ζ65ζ65ζ6ζ65ζ6-1ζ6-11ζ32ζ3    linear of order 6
ρ411ζ32ζ311ζ32ζ3ζ3ζ32ζ3ζ3ζ32ζ3ζ321ζ3211ζ32ζ3    linear of order 3
ρ51-1ζ32ζ3ζ32ζ3ζ3ζ3211-1ζ65ζ6ζ6-1ζ65ζ65ζ61ζ32ζ3    linear of order 6
ρ61-111ζ3ζ32ζ3ζ32ζ3ζ32ζ65-1-1ζ6ζ6ζ6ζ65ζ65111    linear of order 6
ρ71-1ζ32ζ3ζ3ζ3211ζ32ζ3ζ6ζ65ζ6-1ζ65ζ6-1ζ651ζ32ζ3    linear of order 6
ρ81111ζ3ζ32ζ3ζ32ζ3ζ32ζ311ζ32ζ32ζ32ζ3ζ3111    linear of order 3
ρ911ζ3ζ32ζ32ζ311ζ3ζ32ζ3ζ32ζ31ζ32ζ31ζ321ζ3ζ32    linear of order 3
ρ101111ζ32ζ3ζ32ζ3ζ32ζ3ζ3211ζ3ζ3ζ3ζ32ζ32111    linear of order 3
ρ1111ζ32ζ3ζ32ζ3ζ3ζ32111ζ3ζ32ζ321ζ3ζ3ζ321ζ32ζ3    linear of order 3
ρ1211ζ3ζ32ζ3ζ32ζ32ζ3111ζ32ζ3ζ31ζ32ζ32ζ31ζ3ζ32    linear of order 3
ρ1311ζ3ζ3211ζ3ζ32ζ32ζ3ζ32ζ32ζ3ζ32ζ31ζ311ζ3ζ32    linear of order 3
ρ141-1ζ3ζ3211ζ3ζ32ζ32ζ3ζ6ζ6ζ65ζ6ζ65-1ζ65-11ζ3ζ32    linear of order 6
ρ151-1ζ3ζ32ζ32ζ311ζ3ζ32ζ65ζ6ζ65-1ζ6ζ65-1ζ61ζ3ζ32    linear of order 6
ρ161-1ζ3ζ32ζ3ζ32ζ32ζ311-1ζ6ζ65ζ65-1ζ6ζ6ζ651ζ3ζ32    linear of order 6
ρ171-111ζ32ζ3ζ32ζ3ζ32ζ3ζ6-1-1ζ65ζ65ζ65ζ6ζ6111    linear of order 6
ρ1811ζ32ζ3ζ3ζ3211ζ32ζ3ζ32ζ3ζ321ζ3ζ321ζ31ζ32ζ3    linear of order 3
ρ19606600000000000000-1-1-1    orthogonal lifted from F7
ρ2060-3+3-3-3-3-300000000000000-1ζ65ζ6    complex faithful
ρ2160-3-3-3-3+3-300000000000000-1ζ6ζ65    complex faithful

Permutation representations of C3×F7
On 21 points - transitive group 21T9
Generators in S21
(1 15 8)(2 16 9)(3 17 10)(4 18 11)(5 19 12)(6 20 13)(7 21 14)
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)
(1 15 8)(2 18 10 7 19 13)(3 21 12 6 16 11)(4 17 14 5 20 9)

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

G:=Group( (1,15,8)(2,16,9)(3,17,10)(4,18,11)(5,19,12)(6,20,13)(7,21,14), (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21), (1,15,8)(2,18,10,7,19,13)(3,21,12,6,16,11)(4,17,14,5,20,9) );

G=PermutationGroup([(1,15,8),(2,16,9),(3,17,10),(4,18,11),(5,19,12),(6,20,13),(7,21,14)], [(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21)], [(1,15,8),(2,18,10,7,19,13),(3,21,12,6,16,11),(4,17,14,5,20,9)])

G:=TransitiveGroup(21,9);

C3×F7 is a maximal subgroup of   C93F7  C94F7  D7⋊He3
C3×F7 is a maximal quotient of   C93F7  C94F7  C32.F7  D7⋊He3

Matrix representation of C3×F7 in GL7(𝔽43)

6000000
0100000
0010000
0001000
0000100
0000010
0000001
,
1000000
00000042
01000042
00100042
00010042
00001042
00000142
,
42000000
0000010
0001000
0100000
0000001
0000100
0010000

G:=sub<GL(7,GF(43))| [6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,42,42,42,42,42,42],[42,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0] >;

C3×F7 in GAP, Magma, Sage, TeX

C_3\times F_7
% in TeX

G:=Group("C3xF7");
// GroupNames label

G:=SmallGroup(126,7);
// by ID

G=gap.SmallGroup(126,7);
# by ID

G:=PCGroup([4,-2,-3,-3,-7,1731,583]);
// Polycyclic

G:=Group<a,b,c|a^3=b^7=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations

Export

Subgroup lattice of C3×F7 in TeX
Character table of C3×F7 in TeX

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