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G = C6×C7⋊C3order 126 = 2·32·7

Direct product of C6 and C7⋊C3

direct product, metacyclic, supersoluble, monomial, A-group, 3-hyperelementary

Aliases: C6×C7⋊C3, C42⋊C3, C14⋊C32, C214C6, C72(C3×C6), SmallGroup(126,10)

Series: Derived Chief Lower central Upper central

C1C7 — C6×C7⋊C3
C1C7C21C3×C7⋊C3 — C6×C7⋊C3
C7 — C6×C7⋊C3
C1C6

Generators and relations for C6×C7⋊C3
 G = < a,b,c | a6=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >

7C3
7C3
7C3
7C6
7C6
7C6
7C32
7C3×C6

Character table of C6×C7⋊C3

 class 123A3B3C3D3E3F3G3H6A6B6C6D6E6F6G6H7A7B14A14B21A21B21C21D42A42B42C42D
 size 111177777711777777333333333333
ρ1111111111111111111111111111111    trivial
ρ21-111111111-1-1-1-1-1-1-1-111-1-11111-1-1-1-1    linear of order 2
ρ31-1ζ3ζ321ζ31ζ32ζ32ζ3ζ65ζ6-1ζ65-1ζ6ζ6ζ6511-1-1ζ3ζ3ζ32ζ32ζ6ζ65ζ65ζ6    linear of order 6
ρ41111ζ3ζ32ζ32ζ32ζ3ζ311ζ3ζ32ζ32ζ32ζ3ζ3111111111111    linear of order 3
ρ51-111ζ3ζ32ζ32ζ32ζ3ζ3-1-1ζ65ζ6ζ6ζ6ζ65ζ6511-1-11111-1-1-1-1    linear of order 6
ρ611ζ32ζ3ζ321ζ3ζ321ζ3ζ32ζ3ζ321ζ3ζ321ζ31111ζ32ζ32ζ3ζ3ζ3ζ32ζ32ζ3    linear of order 3
ρ711ζ3ζ32ζ32ζ32ζ31ζ31ζ3ζ32ζ32ζ32ζ31ζ311111ζ3ζ3ζ32ζ32ζ32ζ3ζ3ζ32    linear of order 3
ρ81-1ζ32ζ3ζ321ζ3ζ321ζ3ζ6ζ65ζ6-1ζ65ζ6-1ζ6511-1-1ζ32ζ32ζ3ζ3ζ65ζ6ζ6ζ65    linear of order 6
ρ911ζ3ζ32ζ31ζ32ζ31ζ32ζ3ζ32ζ31ζ32ζ31ζ321111ζ3ζ3ζ32ζ32ζ32ζ3ζ3ζ32    linear of order 3
ρ1011ζ3ζ321ζ31ζ32ζ32ζ3ζ3ζ321ζ31ζ32ζ32ζ31111ζ3ζ3ζ32ζ32ζ32ζ3ζ3ζ32    linear of order 3
ρ111-1ζ3ζ32ζ32ζ32ζ31ζ31ζ65ζ6ζ6ζ6ζ65-1ζ65-111-1-1ζ3ζ3ζ32ζ32ζ6ζ65ζ65ζ6    linear of order 6
ρ1211ζ32ζ3ζ3ζ3ζ321ζ321ζ32ζ3ζ3ζ3ζ321ζ3211111ζ32ζ32ζ3ζ3ζ3ζ32ζ32ζ3    linear of order 3
ρ1311ζ32ζ31ζ321ζ3ζ3ζ32ζ32ζ31ζ321ζ3ζ3ζ321111ζ32ζ32ζ3ζ3ζ3ζ32ζ32ζ3    linear of order 3
ρ141-111ζ32ζ3ζ3ζ3ζ32ζ32-1-1ζ6ζ65ζ65ζ65ζ6ζ611-1-11111-1-1-1-1    linear of order 6
ρ151-1ζ32ζ3ζ3ζ3ζ321ζ321ζ6ζ65ζ65ζ65ζ6-1ζ6-111-1-1ζ32ζ32ζ3ζ3ζ65ζ6ζ6ζ65    linear of order 6
ρ161111ζ32ζ3ζ3ζ3ζ32ζ3211ζ32ζ3ζ3ζ3ζ32ζ32111111111111    linear of order 3
ρ171-1ζ3ζ32ζ31ζ32ζ31ζ32ζ65ζ6ζ65-1ζ6ζ65-1ζ611-1-1ζ3ζ3ζ32ζ32ζ6ζ65ζ65ζ6    linear of order 6
ρ181-1ζ32ζ31ζ321ζ3ζ3ζ32ζ6ζ65-1ζ6-1ζ65ζ65ζ611-1-1ζ32ζ32ζ3ζ3ζ65ζ6ζ6ζ65    linear of order 6
ρ19333300000033000000-1+-7/2-1--7/2-1+-7/2-1--7/2-1--7/2-1+-7/2-1+-7/2-1--7/2-1+-7/2-1+-7/2-1--7/2-1--7/2    complex lifted from C7⋊C3
ρ203-333000000-3-3000000-1--7/2-1+-7/21+-7/21--7/2-1+-7/2-1--7/2-1--7/2-1+-7/21+-7/21+-7/21--7/21--7/2    complex lifted from C2×C7⋊C3
ρ21333300000033000000-1--7/2-1+-7/2-1--7/2-1+-7/2-1+-7/2-1--7/2-1--7/2-1+-7/2-1--7/2-1--7/2-1+-7/2-1+-7/2    complex lifted from C7⋊C3
ρ223-333000000-3-3000000-1+-7/2-1--7/21--7/21+-7/2-1--7/2-1+-7/2-1+-7/2-1--7/21--7/21--7/21+-7/21+-7/2    complex lifted from C2×C7⋊C3
ρ233-3-3-3-3/2-3+3-3/20000003+3-3/23-3-3/2000000-1+-7/2-1--7/21--7/21+-7/2ζ32ζ7632ζ7532ζ73ζ32ζ7432ζ7232ζ7ζ3ζ743ζ723ζ7ζ3ζ763ζ753ζ733ζ743ζ723ζ732ζ7432ζ7232ζ732ζ7632ζ7532ζ733ζ763ζ753ζ73    complex faithful
ρ243-3-3+3-3/2-3-3-3/20000003-3-3/23+3-3/2000000-1--7/2-1+-7/21+-7/21--7/2ζ3ζ743ζ723ζ7ζ3ζ763ζ753ζ73ζ32ζ7632ζ7532ζ73ζ32ζ7432ζ7232ζ732ζ7632ζ7532ζ733ζ763ζ753ζ733ζ743ζ723ζ732ζ7432ζ7232ζ7    complex faithful
ρ2533-3+3-3/2-3-3-3/2000000-3+3-3/2-3-3-3/2000000-1+-7/2-1--7/2-1+-7/2-1--7/2ζ3ζ763ζ753ζ73ζ3ζ743ζ723ζ7ζ32ζ7432ζ7232ζ7ζ32ζ7632ζ7532ζ73ζ32ζ7432ζ7232ζ7ζ3ζ743ζ723ζ7ζ3ζ763ζ753ζ73ζ32ζ7632ζ7532ζ73    complex lifted from C3×C7⋊C3
ρ2633-3+3-3/2-3-3-3/2000000-3+3-3/2-3-3-3/2000000-1--7/2-1+-7/2-1--7/2-1+-7/2ζ3ζ743ζ723ζ7ζ3ζ763ζ753ζ73ζ32ζ7632ζ7532ζ73ζ32ζ7432ζ7232ζ7ζ32ζ7632ζ7532ζ73ζ3ζ763ζ753ζ73ζ3ζ743ζ723ζ7ζ32ζ7432ζ7232ζ7    complex lifted from C3×C7⋊C3
ρ2733-3-3-3/2-3+3-3/2000000-3-3-3/2-3+3-3/2000000-1+-7/2-1--7/2-1+-7/2-1--7/2ζ32ζ7632ζ7532ζ73ζ32ζ7432ζ7232ζ7ζ3ζ743ζ723ζ7ζ3ζ763ζ753ζ73ζ3ζ743ζ723ζ7ζ32ζ7432ζ7232ζ7ζ32ζ7632ζ7532ζ73ζ3ζ763ζ753ζ73    complex lifted from C3×C7⋊C3
ρ283-3-3-3-3/2-3+3-3/20000003+3-3/23-3-3/2000000-1--7/2-1+-7/21+-7/21--7/2ζ32ζ7432ζ7232ζ7ζ32ζ7632ζ7532ζ73ζ3ζ763ζ753ζ73ζ3ζ743ζ723ζ73ζ763ζ753ζ7332ζ7632ζ7532ζ7332ζ7432ζ7232ζ73ζ743ζ723ζ7    complex faithful
ρ2933-3-3-3/2-3+3-3/2000000-3-3-3/2-3+3-3/2000000-1--7/2-1+-7/2-1--7/2-1+-7/2ζ32ζ7432ζ7232ζ7ζ32ζ7632ζ7532ζ73ζ3ζ763ζ753ζ73ζ3ζ743ζ723ζ7ζ3ζ763ζ753ζ73ζ32ζ7632ζ7532ζ73ζ32ζ7432ζ7232ζ7ζ3ζ743ζ723ζ7    complex lifted from C3×C7⋊C3
ρ303-3-3+3-3/2-3-3-3/20000003-3-3/23+3-3/2000000-1+-7/2-1--7/21--7/21+-7/2ζ3ζ763ζ753ζ73ζ3ζ743ζ723ζ7ζ32ζ7432ζ7232ζ7ζ32ζ7632ζ7532ζ7332ζ7432ζ7232ζ73ζ743ζ723ζ73ζ763ζ753ζ7332ζ7632ζ7532ζ73    complex faithful

Smallest permutation representation of C6×C7⋊C3
On 42 points
Generators in S42
(1 29 15 22 8 36)(2 30 16 23 9 37)(3 31 17 24 10 38)(4 32 18 25 11 39)(5 33 19 26 12 40)(6 34 20 27 13 41)(7 35 21 28 14 42)
(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)
(1 15 8)(2 17 12)(3 19 9)(4 21 13)(5 16 10)(6 18 14)(7 20 11)(22 36 29)(23 38 33)(24 40 30)(25 42 34)(26 37 31)(27 39 35)(28 41 32)

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

G:=Group( (1,29,15,22,8,36)(2,30,16,23,9,37)(3,31,17,24,10,38)(4,32,18,25,11,39)(5,33,19,26,12,40)(6,34,20,27,13,41)(7,35,21,28,14,42), (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), (1,15,8)(2,17,12)(3,19,9)(4,21,13)(5,16,10)(6,18,14)(7,20,11)(22,36,29)(23,38,33)(24,40,30)(25,42,34)(26,37,31)(27,39,35)(28,41,32) );

G=PermutationGroup([(1,29,15,22,8,36),(2,30,16,23,9,37),(3,31,17,24,10,38),(4,32,18,25,11,39),(5,33,19,26,12,40),(6,34,20,27,13,41),(7,35,21,28,14,42)], [(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)], [(1,15,8),(2,17,12),(3,19,9),(4,21,13),(5,16,10),(6,18,14),(7,20,11),(22,36,29),(23,38,33),(24,40,30),(25,42,34),(26,37,31),(27,39,35),(28,41,32)])

C6×C7⋊C3 is a maximal subgroup of   C6.F7

Matrix representation of C6×C7⋊C3 in GL3(𝔽43) generated by

3700
0370
0037
,
24251
100
010
,
3600
377
0360
G:=sub<GL(3,GF(43))| [37,0,0,0,37,0,0,0,37],[24,1,0,25,0,1,1,0,0],[36,3,0,0,7,36,0,7,0] >;

C6×C7⋊C3 in GAP, Magma, Sage, TeX

C_6\times C_7\rtimes C_3
% in TeX

G:=Group("C6xC7:C3");
// GroupNames label

G:=SmallGroup(126,10);
// by ID

G=gap.SmallGroup(126,10);
# by ID

G:=PCGroup([4,-2,-3,-3,-7,295]);
// Polycyclic

G:=Group<a,b,c|a^6=b^7=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

Export

Subgroup lattice of C6×C7⋊C3 in TeX
Character table of C6×C7⋊C3 in TeX

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