direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C14×C7⋊C3, C14⋊C21, C7⋊2C42, C72⋊8C6, (C7×C14)⋊1C3, SmallGroup(294,15)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C14×C7⋊C3 |
Generators and relations for C14×C7⋊C3
G = < a,b,c | a14=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >
Smallest permutation representation of C14×C7⋊C3
►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 3 5 7 9 11 13)(2 4 6 8 10 12 14)(15 19 23 27 17 21 25)(16 20 24 28 18 22 26)(29 37 31 39 33 41 35)(30 38 32 40 34 42 36)
(1 30 16)(2 31 17)(3 32 18)(4 33 19)(5 34 20)(6 35 21)(7 36 22)(8 37 23)(9 38 24)(10 39 25)(11 40 26)(12 41 27)(13 42 28)(14 29 15)
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), (1,3,5,7,9,11,13)(2,4,6,8,10,12,14)(15,19,23,27,17,21,25)(16,20,24,28,18,22,26)(29,37,31,39,33,41,35)(30,38,32,40,34,42,36), (1,30,16)(2,31,17)(3,32,18)(4,33,19)(5,34,20)(6,35,21)(7,36,22)(8,37,23)(9,38,24)(10,39,25)(11,40,26)(12,41,27)(13,42,28)(14,29,15)>;
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), (1,3,5,7,9,11,13)(2,4,6,8,10,12,14)(15,19,23,27,17,21,25)(16,20,24,28,18,22,26)(29,37,31,39,33,41,35)(30,38,32,40,34,42,36), (1,30,16)(2,31,17)(3,32,18)(4,33,19)(5,34,20)(6,35,21)(7,36,22)(8,37,23)(9,38,24)(10,39,25)(11,40,26)(12,41,27)(13,42,28)(14,29,15) );
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)], [(1,3,5,7,9,11,13),(2,4,6,8,10,12,14),(15,19,23,27,17,21,25),(16,20,24,28,18,22,26),(29,37,31,39,33,41,35),(30,38,32,40,34,42,36)], [(1,30,16),(2,31,17),(3,32,18),(4,33,19),(5,34,20),(6,35,21),(7,36,22),(8,37,23),(9,38,24),(10,39,25),(11,40,26),(12,41,27),(13,42,28),(14,29,15)]])
70 conjugacy classes
| class | 1 | 2 | 3A | 3B | 6A | 6B | 7A | ··· | 7F | 7G | ··· | 7T | 14A | ··· | 14F | 14G | ··· | 14T | 21A | ··· | 21L | 42A | ··· | 42L |
| order | 1 | 2 | 3 | 3 | 6 | 6 | 7 | ··· | 7 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 21 | ··· | 21 | 42 | ··· | 42 |
| size | 1 | 1 | 7 | 7 | 7 | 7 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 7 | ··· | 7 | 7 | ··· | 7 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 |
| type | + | + | ||||||||||
| image | C1 | C2 | C3 | C6 | C7 | C14 | C21 | C42 | C7⋊C3 | C2×C7⋊C3 | C7×C7⋊C3 | C14×C7⋊C3 |
| kernel | C14×C7⋊C3 | C7×C7⋊C3 | C7×C14 | C72 | C2×C7⋊C3 | C7⋊C3 | C14 | C7 | C14 | C7 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 12 | 12 | 2 | 2 | 12 | 12 |
Matrix representation of C14×C7⋊C3 ►in GL3(𝔽43) generated by
| 22 | 0 | 0 |
| 0 | 22 | 0 |
| 0 | 0 | 22 |
| 11 | 9 | 6 |
| 0 | 35 | 0 |
| 0 | 0 | 21 |
| 6 | 0 | 0 |
| 0 | 0 | 1 |
| 10 | 7 | 37 |
G:=sub<GL(3,GF(43))| [22,0,0,0,22,0,0,0,22],[11,0,0,9,35,0,6,0,21],[6,0,10,0,0,7,0,1,37] >;
C14×C7⋊C3 in GAP, Magma, Sage, TeX
C_{14}\times C_7\rtimes C_3 % in TeX
G:=Group("C14xC7:C3"); // GroupNames label
G:=SmallGroup(294,15);
// by ID
G=gap.SmallGroup(294,15);
# by ID
G:=PCGroup([4,-2,-3,-7,-7,679]);
// Polycyclic
G:=Group<a,b,c|a^14=b^7=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations
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