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G = S3×F7order 252 = 22·32·7

Direct product of S3 and F7

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

Aliases: S3×F7, D21⋊C6, C3⋊F7⋊C2, (S3×C7)⋊C6, C21⋊(C2×C6), C71(S3×C6), (C3×D7)⋊C6, (S3×D7)⋊C3, D7⋊(C3×S3), C7⋊C31D6, (C3×F7)⋊C2, C31(C2×F7), (S3×C7⋊C3)⋊C2, (C3×C7⋊C3)⋊C22, Aut(D21), Hol(C21), SmallGroup(252,26)

Series: Derived Chief Lower central Upper central

C1C21 — S3×F7
C1C7C21C3×C7⋊C3C3×F7 — S3×F7
C21 — S3×F7
C1

Generators and relations for S3×F7
 G = < a,b,c,d | a3=b2=c7=d6=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >

3C2
7C2
21C2
7C3
14C3
21C22
7C6
7C6
7S3
14C6
21C6
21C6
7C32
3C14
3D7
2C7⋊C3
7D6
21C2×C6
7C3×S3
7C3×S3
7C3×C6
3D14
2F7
3C2×C7⋊C3
3F7
7S3×C6
3C2×F7

Character table of S3×F7

 class 12A2B2C3A3B3C3D3E6A6B6C6D6E6F6G6H6I71421
 size 137212771414771414142121212161812
ρ1111111111111111111111    trivial
ρ21-11-11111111111-1-1-1-11-11    linear of order 2
ρ31-1-1111111-1-1-1-1-11-11-11-11    linear of order 2
ρ411-1-111111-1-1-1-1-1-11-11111    linear of order 2
ρ51-11-11ζ3ζ32ζ32ζ3ζ32ζ3ζ32ζ31ζ65ζ6ζ6ζ651-11    linear of order 6
ρ61-1-111ζ32ζ3ζ3ζ32ζ65ζ6ζ65ζ6-1ζ32ζ65ζ3ζ61-11    linear of order 6
ρ711111ζ3ζ32ζ32ζ3ζ32ζ3ζ32ζ31ζ3ζ32ζ32ζ3111    linear of order 3
ρ81-11-11ζ32ζ3ζ3ζ32ζ3ζ32ζ3ζ321ζ6ζ65ζ65ζ61-11    linear of order 6
ρ911-1-11ζ3ζ32ζ32ζ3ζ6ζ65ζ6ζ65-1ζ65ζ32ζ6ζ3111    linear of order 6
ρ1011111ζ32ζ3ζ3ζ32ζ3ζ32ζ3ζ321ζ32ζ3ζ3ζ32111    linear of order 3
ρ111-1-111ζ3ζ32ζ32ζ3ζ6ζ65ζ6ζ65-1ζ3ζ6ζ32ζ651-11    linear of order 6
ρ1211-1-11ζ32ζ3ζ3ζ32ζ65ζ6ζ65ζ6-1ζ6ζ3ζ65ζ32111    linear of order 6
ρ1320-20-122-1-1-2-2111000020-1    orthogonal lifted from D6
ρ142020-122-1-122-1-1-1000020-1    orthogonal lifted from S3
ρ1520-20-1-1+-3-1--3ζ6ζ651+-31--3ζ32ζ31000020-1    complex lifted from S3×C6
ρ162020-1-1+-3-1--3ζ6ζ65-1--3-1+-3ζ6ζ65-1000020-1    complex lifted from C3×S3
ρ172020-1-1--3-1+-3ζ65ζ6-1+-3-1--3ζ65ζ6-1000020-1    complex lifted from C3×S3
ρ1820-20-1-1--3-1+-3ζ65ζ61--31+-3ζ3ζ321000020-1    complex lifted from S3×C6
ρ196-60060000000000000-11-1    orthogonal lifted from C2×F7
ρ20660060000000000000-1-1-1    orthogonal lifted from F7
ρ2112000-60000000000000-201    orthogonal faithful

Permutation representations of S3×F7
On 21 points - transitive group 21T15
Generators in S21
(1 8 15)(2 9 16)(3 10 17)(4 11 18)(5 12 19)(6 13 20)(7 14 21)
(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)
(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)(9 11 10 14 12 13)(16 18 17 21 19 20)

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

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

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

G:=TransitiveGroup(21,15);

Matrix representation of S3×F7 in GL8(𝔽43)

042000000
142000000
00100000
00010000
00001000
00000100
00000010
00000001
,
01000000
10000000
00100000
00010000
00001000
00000100
00000010
00000001
,
10000000
01000000
000000042
001000042
000100042
000010042
000001042
000000142
,
60000000
06000000
00000010
00001000
00100000
00000001
00000100
00010000

G:=sub<GL(8,GF(43))| [0,1,0,0,0,0,0,0,42,42,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,42,42,42,42,42,42],[6,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0] >;

S3×F7 in GAP, Magma, Sage, TeX

S_3\times F_7
% in TeX

G:=Group("S3xF7");
// GroupNames label

G:=SmallGroup(252,26);
// by ID

G=gap.SmallGroup(252,26);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-7,248,5404,914]);
// Polycyclic

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

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Subgroup lattice of S3×F7 in TeX
Character table of S3×F7 in TeX

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