Copied to
clipboard

G = C7×F7order 294 = 2·3·72

Direct product of C7 and F7

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

Aliases: C7×F7, C7⋊C42, D7⋊C21, C721C6, C7⋊C3⋊C14, (C7×D7)⋊1C3, (C7×C7⋊C3)⋊1C2, SmallGroup(294,8)

Series: Derived Chief Lower central Upper central

C1C7 — C7×F7
C1C7C72C7×C7⋊C3 — C7×F7
C7 — C7×F7
C1C7

Generators and relations for C7×F7
 G = < a,b,c | a7=b7=c6=1, ab=ba, ac=ca, cbc-1=b5 >

7C2
7C3
6C7
7C6
7C14
7C21
7C42

Smallest permutation representation of C7×F7
On 42 points
Generators in S42
(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 2 3 4 5 6 7)(8 12 9 13 10 14 11)(15 17 19 21 16 18 20)(22 28 27 26 25 24 23)(29 32 35 31 34 30 33)(36 41 39 37 42 40 38)
(1 36 8 22 15 29)(2 37 9 23 16 30)(3 38 10 24 17 31)(4 39 11 25 18 32)(5 40 12 26 19 33)(6 41 13 27 20 34)(7 42 14 28 21 35)

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

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

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

49 conjugacy classes

class 1  2 3A3B6A6B7A···7F7G···7M14A···14F21A···21L42A···42L
order1233667···77···714···1421···2142···42
size1777771···16···67···77···77···7

49 irreducible representations

dim1111111166
type+++
imageC1C2C3C6C7C14C21C42F7C7×F7
kernelC7×F7C7×C7⋊C3C7×D7C72F7C7⋊C3D7C7C7C1
# reps112266121216

Matrix representation of C7×F7 in GL6(𝔽43)

400000
040000
004000
000400
000040
000004
,
4019343825
0414239511
0016000
0001100
0000210
0000035
,
26330000
1690000
000100
0336343327
001000
306101117

G:=sub<GL(6,GF(43))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,41,0,0,0,0,19,42,16,0,0,0,34,39,0,11,0,0,38,5,0,0,21,0,25,11,0,0,0,35],[26,16,0,0,0,3,33,9,0,3,0,0,0,0,0,36,1,6,0,0,1,34,0,10,0,0,0,33,0,11,0,0,0,27,0,17] >;

C7×F7 in GAP, Magma, Sage, TeX

C_7\times F_7
% in TeX

G:=Group("C7xF7");
// GroupNames label

G:=SmallGroup(294,8);
// by ID

G=gap.SmallGroup(294,8);
# by ID

G:=PCGroup([4,-2,-3,-7,-7,4035,1351]);
// Polycyclic

G:=Group<a,b,c|a^7=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 C7×F7 in TeX

׿
×
𝔽