direct product, metacyclic, supersoluble, monomial, A-group
Aliases: D7xC28, C28:2C14, D14.C14, Dic7:2C14, C14.18D14, (C7xC28):3C2, C7:1(C2xC28), C72:4(C2xC4), C2.1(D7xC14), C14.2(C2xC14), (C7xDic7):5C2, (D7xC14).2C2, (C7xC14).7C22, SmallGroup(392,24)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — D7xC28 |
Generators and relations for D7xC28
G = < a,b,c | a28=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >
Quotients: C1, C2, C4, C22, C7, C2xC4, D7, C14, C28, D14, C2xC14, C4xD7, C2xC28, C7xD7, D7xC14, D7xC28
Smallest permutation representation of D7xC28
►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 9 17 25 5 13 21)(2 10 18 26 6 14 22)(3 11 19 27 7 15 23)(4 12 20 28 8 16 24)(29 49 41 33 53 45 37)(30 50 42 34 54 46 38)(31 51 43 35 55 47 39)(32 52 44 36 56 48 40)
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 55)(12 56)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 40)(25 41)(26 42)(27 43)(28 44)
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), (1,9,17,25,5,13,21)(2,10,18,26,6,14,22)(3,11,19,27,7,15,23)(4,12,20,28,8,16,24)(29,49,41,33,53,45,37)(30,50,42,34,54,46,38)(31,51,43,35,55,47,39)(32,52,44,36,56,48,40), (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43)(28,44)>;
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), (1,9,17,25,5,13,21)(2,10,18,26,6,14,22)(3,11,19,27,7,15,23)(4,12,20,28,8,16,24)(29,49,41,33,53,45,37)(30,50,42,34,54,46,38)(31,51,43,35,55,47,39)(32,52,44,36,56,48,40), (1,45)(2,46)(3,47)(4,48)(5,49)(6,50)(7,51)(8,52)(9,53)(10,54)(11,55)(12,56)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(25,41)(26,42)(27,43)(28,44) );
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)], [(1,9,17,25,5,13,21),(2,10,18,26,6,14,22),(3,11,19,27,7,15,23),(4,12,20,28,8,16,24),(29,49,41,33,53,45,37),(30,50,42,34,54,46,38),(31,51,43,35,55,47,39),(32,52,44,36,56,48,40)], [(1,45),(2,46),(3,47),(4,48),(5,49),(6,50),(7,51),(8,52),(9,53),(10,54),(11,55),(12,56),(13,29),(14,30),(15,31),(16,32),(17,33),(18,34),(19,35),(20,36),(21,37),(22,38),(23,39),(24,40),(25,41),(26,42),(27,43),(28,44)]])
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 7A | ··· | 7F | 7G | ··· | 7AA | 14A | ··· | 14F | 14G | ··· | 14AA | 14AB | ··· | 14AM | 28A | ··· | 28L | 28M | ··· | 28BB | 28BC | ··· | 28BN |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 7 | 7 | 1 | 1 | 7 | 7 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 7 | ··· | 7 | 1 | ··· | 1 | 2 | ··· | 2 | 7 | ··· | 7 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C7 | C14 | C14 | C14 | C28 | D7 | D14 | C4xD7 | C7xD7 | D7xC14 | D7xC28 |
| kernel | D7xC28 | C7xDic7 | C7xC28 | D7xC14 | C7xD7 | C4xD7 | Dic7 | C28 | D14 | D7 | C28 | C14 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 6 | 6 | 24 | 3 | 3 | 6 | 18 | 18 | 36 |
Matrix representation of D7xC28 ►in GL2(F29) generated by
| 15 | 0 |
| 0 | 15 |
| 19 | 14 |
| 11 | 28 |
| 1 | 0 |
| 11 | 28 |
G:=sub<GL(2,GF(29))| [15,0,0,15],[19,11,14,28],[1,11,0,28] >;
D7xC28 in GAP, Magma, Sage, TeX
D_7\times C_{28} % in TeX
G:=Group("D7xC28"); // GroupNames label
G:=SmallGroup(392,24);
// by ID
G=gap.SmallGroup(392,24);
# by ID
G:=PCGroup([5,-2,-2,-7,-2,-7,146,8404]);
// Polycyclic
G:=Group<a,b,c|a^28=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export