direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C7×F7, C7⋊C42, D7⋊C21, C72⋊1C6, C7⋊C3⋊C14, (C7×D7)⋊1C3, (C7×C7⋊C3)⋊1C2, SmallGroup(294,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C7×F7 |
Generators and relations for C7×F7
G = < a,b,c | a7=b7=c6=1, ab=ba, ac=ca, cbc-1=b5 >
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 | 3A | 3B | 6A | 6B | 7A | ··· | 7F | 7G | ··· | 7M | 14A | ··· | 14F | 21A | ··· | 21L | 42A | ··· | 42L |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 7 | ··· | 7 | 7 | ··· | 7 | 14 | ··· | 14 | 21 | ··· | 21 | 42 | ··· | 42 |
| size | 1 | 7 | 7 | 7 | 7 | 7 | 1 | ··· | 1 | 6 | ··· | 6 | 7 | ··· | 7 | 7 | ··· | 7 | 7 | ··· | 7 |
49 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 |
| type | + | + | + | |||||||
| image | C1 | C2 | C3 | C6 | C7 | C14 | C21 | C42 | F7 | C7×F7 |
| kernel | C7×F7 | C7×C7⋊C3 | C7×D7 | C72 | F7 | C7⋊C3 | D7 | C7 | C7 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 12 | 12 | 1 | 6 |
Matrix representation of C7×F7 ►in GL6(𝔽43)
| 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 | 19 | 34 | 38 | 25 |
| 0 | 41 | 42 | 39 | 5 | 11 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 0 |
| 0 | 0 | 0 | 0 | 0 | 35 |
| 26 | 33 | 0 | 0 | 0 | 0 |
| 16 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 3 | 36 | 34 | 33 | 27 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 3 | 0 | 6 | 10 | 11 | 17 |
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