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G = C2×C3⋊F7order 252 = 22·32·7

Direct product of C2 and C3⋊F7

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

Aliases: C2×C3⋊F7, C6⋊F7, D42⋊C3, C421C6, D212C6, C14⋊(C3×S3), C72(S3×C6), C7⋊C32D6, C32(C2×F7), C212(C2×C6), (C2×C7⋊C3)⋊S3, (C6×C7⋊C3)⋊1C2, (C3×C7⋊C3)⋊2C22, SmallGroup(252,30)

Series: Derived Chief Lower central Upper central

C1C21 — C2×C3⋊F7
C1C7C21C3×C7⋊C3C3⋊F7 — C2×C3⋊F7
C21 — C2×C3⋊F7
C1C2

Generators and relations for C2×C3⋊F7
 G = < a,b,c,d | a2=b3=c7=d6=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c5 >

21C2
21C2
7C3
14C3
21C22
7S3
7C6
7S3
14C6
21C6
21C6
7C32
3D7
3D7
2C7⋊C3
7D6
21C2×C6
7C3×S3
7C3×C6
7C3×S3
3D14
2C2×C7⋊C3
3F7
3F7
7S3×C6
3C2×F7

Character table of C2×C3⋊F7

 class 12A2B2C3A3B3C3D3E6A6B6C6D6E6F6G6H6I71421A21B42A42B
 size 1121212771414277141421212121666666
ρ1111111111111111111111111    trivial
ρ21-11-111111-1-1-1-1-11-11-11-111-1-1    linear of order 2
ρ31-1-1111111-1-1-1-1-1-11-111-111-1-1    linear of order 2
ρ411-1-11111111111-1-1-1-1111111    linear of order 2
ρ51-1-111ζ32ζ3ζ3ζ32-1ζ6ζ65ζ65ζ6ζ6ζ3ζ65ζ321-111-1-1    linear of order 6
ρ611-1-11ζ3ζ32ζ32ζ31ζ3ζ32ζ32ζ3ζ65ζ6ζ6ζ65111111    linear of order 6
ρ71-11-11ζ3ζ32ζ32ζ3-1ζ65ζ6ζ6ζ65ζ3ζ6ζ32ζ651-111-1-1    linear of order 6
ρ811111ζ32ζ3ζ3ζ321ζ32ζ3ζ3ζ32ζ32ζ3ζ3ζ32111111    linear of order 3
ρ91-11-11ζ32ζ3ζ3ζ32-1ζ6ζ65ζ65ζ6ζ32ζ65ζ3ζ61-111-1-1    linear of order 6
ρ101-1-111ζ3ζ32ζ32ζ3-1ζ65ζ6ζ6ζ65ζ65ζ32ζ6ζ31-111-1-1    linear of order 6
ρ1111111ζ3ζ32ζ32ζ31ζ3ζ32ζ32ζ3ζ3ζ32ζ32ζ3111111    linear of order 3
ρ1211-1-11ζ32ζ3ζ3ζ321ζ32ζ3ζ3ζ32ζ6ζ65ζ65ζ6111111    linear of order 6
ρ132200-122-1-1-122-1-1000022-1-1-1-1    orthogonal lifted from S3
ρ142-200-122-1-11-2-21100002-2-1-111    orthogonal lifted from D6
ρ152200-1-1+-3-1--3ζ6ζ65-1-1+-3-1--3ζ6ζ65000022-1-1-1-1    complex lifted from C3×S3
ρ162200-1-1--3-1+-3ζ65ζ6-1-1--3-1+-3ζ65ζ6000022-1-1-1-1    complex lifted from C3×S3
ρ172-200-1-1--3-1+-3ζ65ζ611+-31--3ζ3ζ3200002-2-1-111    complex lifted from S3×C6
ρ182-200-1-1+-3-1--3ζ6ζ6511--31+-3ζ32ζ300002-2-1-111    complex lifted from S3×C6
ρ196-60060000-600000000-11-1-111    orthogonal lifted from C2×F7
ρ20660060000600000000-1-1-1-1-1-1    orthogonal lifted from F7
ρ216600-30000-300000000-1-11+21/21-21/21+21/21-21/2    orthogonal lifted from C3⋊F7
ρ226-600-30000300000000-111-21/21+21/2-1+21/2-1-21/2    orthogonal faithful
ρ236600-30000-300000000-1-11-21/21+21/21-21/21+21/2    orthogonal lifted from C3⋊F7
ρ246-600-30000300000000-111+21/21-21/2-1-21/2-1+21/2    orthogonal faithful

Smallest permutation representation of C2×C3⋊F7
On 42 points
Generators in S42
(1 27)(2 28)(3 22)(4 23)(5 24)(6 25)(7 26)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)
(1 20 13)(2 21 14)(3 15 8)(4 16 9)(5 17 10)(6 18 11)(7 19 12)(22 36 29)(23 37 30)(24 38 31)(25 39 32)(26 40 33)(27 41 34)(28 42 35)
(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 27)(2 23 3 26 5 25)(4 22 7 24 6 28)(8 40 10 39 14 37)(9 36 12 38 11 42)(13 41)(15 33 17 32 21 30)(16 29 19 31 18 35)(20 34)

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

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

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

Matrix representation of C2×C3⋊F7 in GL8(𝔽43)

420000000
042000000
00100000
00010000
00001000
00000100
00000010
00000001
,
421000000
420000000
0040053850
0038405005
0038382050
0003804055
0038003825
0003853802
,
10000000
01000000
004210000
004201000
004200100
004200010
004200001
004200000
,
042000000
420000000
00000010
00001000
00100000
00000001
00000100
00010000

G:=sub<GL(8,GF(43))| [42,0,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[42,42,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,38,38,0,38,0,0,0,0,40,38,38,0,38,0,0,5,5,2,0,0,5,0,0,38,0,0,40,38,38,0,0,5,0,5,5,2,0,0,0,0,5,0,5,5,2],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,42,42,42,42,42,42,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[0,42,0,0,0,0,0,0,42,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0] >;

C2×C3⋊F7 in GAP, Magma, Sage, TeX

C_2\times C_3\rtimes F_7
% in TeX

G:=Group("C2xC3:F7");
// GroupNames label

G:=SmallGroup(252,30);
// by ID

G=gap.SmallGroup(252,30);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-7,483,5404,464]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^3=c^7=d^6=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations

Export

Subgroup lattice of C2×C3⋊F7 in TeX
Character table of C2×C3⋊F7 in TeX

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