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G = C3⋊F7order 126 = 2·32·7

The semidirect product of C3 and F7 acting via F7/C7⋊C3=C2

metacyclic, supersoluble, monomial, A-group

Aliases: C3⋊F7, D21⋊C3, C211C6, C7⋊C3⋊S3, C7⋊(C3×S3), (C3×C7⋊C3)⋊1C2, SmallGroup(126,9)

Series: Derived Chief Lower central Upper central

C1C21 — C3⋊F7
C1C7C21C3×C7⋊C3 — C3⋊F7
C21 — C3⋊F7
C1

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

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

Character table of C3⋊F7

 class 123A3B3C3D3E6A6B721A21B
 size 12127714142121666
ρ1111111111111    trivial
ρ21-111111-1-1111    linear of order 2
ρ3111ζ3ζ32ζ3ζ32ζ3ζ32111    linear of order 3
ρ4111ζ32ζ3ζ32ζ3ζ32ζ3111    linear of order 3
ρ51-11ζ3ζ32ζ3ζ32ζ65ζ6111    linear of order 6
ρ61-11ζ32ζ3ζ32ζ3ζ6ζ65111    linear of order 6
ρ720-122-1-1002-1-1    orthogonal lifted from S3
ρ820-1-1--3-1+-3ζ6ζ65002-1-1    complex lifted from C3×S3
ρ920-1-1+-3-1--3ζ65ζ6002-1-1    complex lifted from C3×S3
ρ10606000000-1-1-1    orthogonal lifted from F7
ρ1160-3000000-11+21/21-21/2    orthogonal faithful
ρ1260-3000000-11-21/21+21/2    orthogonal faithful

Permutation representations of C3⋊F7
On 21 points - transitive group 21T10
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)
(2 4 3 7 5 6)(8 15)(9 18 10 21 12 20)(11 17 14 19 13 16)

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

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

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

G:=TransitiveGroup(21,10);

C3⋊F7 is a maximal subgroup of   S3×F7  C9⋊F7  C92F7  C95F7  C32⋊F7  C324F7
C3⋊F7 is a maximal quotient of   C6.F7  C9⋊F7  C92F7  C95F7  D21⋊C9  C32⋊F7  C324F7

Matrix representation of C3⋊F7 in GL6(𝔽43)

4038380380
0403838038
552005
3800403838
505520
050552
,
424242424242
100000
010000
001000
000100
000010
,
100000
000001
000100
010000
424242424242
000010

G:=sub<GL(6,GF(43))| [40,0,5,38,5,0,38,40,5,0,0,5,38,38,2,0,5,0,0,38,0,40,5,5,38,0,0,38,2,5,0,38,5,38,0,2],[42,1,0,0,0,0,42,0,1,0,0,0,42,0,0,1,0,0,42,0,0,0,1,0,42,0,0,0,0,1,42,0,0,0,0,0],[1,0,0,0,42,0,0,0,0,1,42,0,0,0,0,0,42,0,0,0,1,0,42,0,0,0,0,0,42,1,0,1,0,0,42,0] >;

C3⋊F7 in GAP, Magma, Sage, TeX

C_3\rtimes F_7
% in TeX

G:=Group("C3:F7");
// GroupNames label

G:=SmallGroup(126,9);
// by ID

G=gap.SmallGroup(126,9);
# by ID

G:=PCGroup([4,-2,-3,-3,-7,146,1731,295]);
// Polycyclic

G:=Group<a,b,c|a^3=b^7=c^6=1,a*b=b*a,c*a*c^-1=a^-1,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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