direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C14×Dic7, C14⋊C28, C14.20D14, C142.1C2, C7⋊2(C2×C28), (C7×C14)⋊2C4, C72⋊6(C2×C4), C22.(C7×D7), (C2×C14).5D7, C2.2(D7×C14), (C2×C14).2C14, C14.4(C2×C14), (C7×C14).9C22, SmallGroup(392,26)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C14×Dic7 |
Generators and relations for C14×Dic7
G = < a,b,c | a14=b14=1, c2=b7, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C14×Dic7
►On 56 points
Generators in S56
(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)(43 44 45 46 47 48 49 50 51 52 53 54 55 56)
(1 44 9 52 3 46 11 54 5 48 13 56 7 50)(2 45 10 53 4 47 12 55 6 49 14 43 8 51)(15 32 21 38 27 30 19 36 25 42 17 34 23 40)(16 33 22 39 28 31 20 37 26 29 18 35 24 41)
(1 16 54 37)(2 17 55 38)(3 18 56 39)(4 19 43 40)(5 20 44 41)(6 21 45 42)(7 22 46 29)(8 23 47 30)(9 24 48 31)(10 25 49 32)(11 26 50 33)(12 27 51 34)(13 28 52 35)(14 15 53 36)
G:=sub<Sym(56)| (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)(43,44,45,46,47,48,49,50,51,52,53,54,55,56), (1,44,9,52,3,46,11,54,5,48,13,56,7,50)(2,45,10,53,4,47,12,55,6,49,14,43,8,51)(15,32,21,38,27,30,19,36,25,42,17,34,23,40)(16,33,22,39,28,31,20,37,26,29,18,35,24,41), (1,16,54,37)(2,17,55,38)(3,18,56,39)(4,19,43,40)(5,20,44,41)(6,21,45,42)(7,22,46,29)(8,23,47,30)(9,24,48,31)(10,25,49,32)(11,26,50,33)(12,27,51,34)(13,28,52,35)(14,15,53,36)>;
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)(43,44,45,46,47,48,49,50,51,52,53,54,55,56), (1,44,9,52,3,46,11,54,5,48,13,56,7,50)(2,45,10,53,4,47,12,55,6,49,14,43,8,51)(15,32,21,38,27,30,19,36,25,42,17,34,23,40)(16,33,22,39,28,31,20,37,26,29,18,35,24,41), (1,16,54,37)(2,17,55,38)(3,18,56,39)(4,19,43,40)(5,20,44,41)(6,21,45,42)(7,22,46,29)(8,23,47,30)(9,24,48,31)(10,25,49,32)(11,26,50,33)(12,27,51,34)(13,28,52,35)(14,15,53,36) );
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),(43,44,45,46,47,48,49,50,51,52,53,54,55,56)], [(1,44,9,52,3,46,11,54,5,48,13,56,7,50),(2,45,10,53,4,47,12,55,6,49,14,43,8,51),(15,32,21,38,27,30,19,36,25,42,17,34,23,40),(16,33,22,39,28,31,20,37,26,29,18,35,24,41)], [(1,16,54,37),(2,17,55,38),(3,18,56,39),(4,19,43,40),(5,20,44,41),(6,21,45,42),(7,22,46,29),(8,23,47,30),(9,24,48,31),(10,25,49,32),(11,26,50,33),(12,27,51,34),(13,28,52,35),(14,15,53,36)]])
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 7A | ··· | 7F | 7G | ··· | 7AA | 14A | ··· | 14R | 14S | ··· | 14CC | 28A | ··· | 28X |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 7 | 7 | 7 | 7 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 7 | ··· | 7 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | ||||||||
| image | C1 | C2 | C2 | C4 | C7 | C14 | C14 | C28 | D7 | Dic7 | D14 | C7×D7 | C7×Dic7 | D7×C14 |
| kernel | C14×Dic7 | C7×Dic7 | C142 | C7×C14 | C2×Dic7 | Dic7 | C2×C14 | C14 | C2×C14 | C14 | C14 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 6 | 12 | 6 | 24 | 3 | 6 | 3 | 18 | 36 | 18 |
Matrix representation of C14×Dic7 ►in GL3(𝔽29) generated by
| 20 | 0 | 0 |
| 0 | 22 | 0 |
| 0 | 0 | 22 |
| 28 | 0 | 0 |
| 0 | 23 | 0 |
| 0 | 0 | 24 |
| 17 | 0 | 0 |
| 0 | 0 | 28 |
| 0 | 28 | 0 |
G:=sub<GL(3,GF(29))| [20,0,0,0,22,0,0,0,22],[28,0,0,0,23,0,0,0,24],[17,0,0,0,0,28,0,28,0] >;
C14×Dic7 in GAP, Magma, Sage, TeX
C_{14}\times {\rm Dic}_7 % in TeX
G:=Group("C14xDic7"); // GroupNames label
G:=SmallGroup(392,26);
// by ID
G=gap.SmallGroup(392,26);
# by ID
G:=PCGroup([5,-2,-2,-7,-2,-7,140,8404]);
// Polycyclic
G:=Group<a,b,c|a^14=b^14=1,c^2=b^7,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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