direct product, metabelian, supersoluble, monomial, A-group
Aliases: D7×Dic7, D14.D7, C14.1D14, (C7×D7)⋊C4, C2.1D72, C7⋊3(C4×D7), (D7×C14).C2, C72⋊2(C2×C4), C7⋊Dic7⋊1C2, C7⋊1(C2×Dic7), (C7×Dic7)⋊2C2, (C7×C14).1C22, SmallGroup(392,18)
Series: Derived ►Chief ►Lower central ►Upper central
| C72 — D7×Dic7 |
Generators and relations for D7×Dic7
G = < a,b,c,d | a7=b2=c14=1, d2=c7, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
7C2
7C2
2C7
2C7
2C7
7C4
7C22
49C4
2C14
2C14
2C14
7C14
7C14
49C2×C4
7C28
7Dic7
7Dic7
14Dic7
14Dic7
14Dic7
Smallest permutation representation of D7×Dic7
►On 56 points
Generators in S56
(1 3 5 7 9 11 13)(2 4 6 8 10 12 14)(15 27 25 23 21 19 17)(16 28 26 24 22 20 18)(29 41 39 37 35 33 31)(30 42 40 38 36 34 32)(43 45 47 49 51 53 55)(44 46 48 50 52 54 56)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 29)(13 30)(14 31)(15 53)(16 54)(17 55)(18 56)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)(27 51)(28 52)
(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 21 8 28)(2 20 9 27)(3 19 10 26)(4 18 11 25)(5 17 12 24)(6 16 13 23)(7 15 14 22)(29 48 36 55)(30 47 37 54)(31 46 38 53)(32 45 39 52)(33 44 40 51)(34 43 41 50)(35 56 42 49)
G:=sub<Sym(56)| (1,3,5,7,9,11,13)(2,4,6,8,10,12,14)(15,27,25,23,21,19,17)(16,28,26,24,22,20,18)(29,41,39,37,35,33,31)(30,42,40,38,36,34,32)(43,45,47,49,51,53,55)(44,46,48,50,52,54,56), (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,29)(13,30)(14,31)(15,53)(16,54)(17,55)(18,56)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(27,51)(28,52), (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), (1,21,8,28)(2,20,9,27)(3,19,10,26)(4,18,11,25)(5,17,12,24)(6,16,13,23)(7,15,14,22)(29,48,36,55)(30,47,37,54)(31,46,38,53)(32,45,39,52)(33,44,40,51)(34,43,41,50)(35,56,42,49)>;
G:=Group( (1,3,5,7,9,11,13)(2,4,6,8,10,12,14)(15,27,25,23,21,19,17)(16,28,26,24,22,20,18)(29,41,39,37,35,33,31)(30,42,40,38,36,34,32)(43,45,47,49,51,53,55)(44,46,48,50,52,54,56), (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,29)(13,30)(14,31)(15,53)(16,54)(17,55)(18,56)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,49)(26,50)(27,51)(28,52), (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), (1,21,8,28)(2,20,9,27)(3,19,10,26)(4,18,11,25)(5,17,12,24)(6,16,13,23)(7,15,14,22)(29,48,36,55)(30,47,37,54)(31,46,38,53)(32,45,39,52)(33,44,40,51)(34,43,41,50)(35,56,42,49) );
G=PermutationGroup([[(1,3,5,7,9,11,13),(2,4,6,8,10,12,14),(15,27,25,23,21,19,17),(16,28,26,24,22,20,18),(29,41,39,37,35,33,31),(30,42,40,38,36,34,32),(43,45,47,49,51,53,55),(44,46,48,50,52,54,56)], [(1,32),(2,33),(3,34),(4,35),(5,36),(6,37),(7,38),(8,39),(9,40),(10,41),(11,42),(12,29),(13,30),(14,31),(15,53),(16,54),(17,55),(18,56),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,49),(26,50),(27,51),(28,52)], [(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)], [(1,21,8,28),(2,20,9,27),(3,19,10,26),(4,18,11,25),(5,17,12,24),(6,16,13,23),(7,15,14,22),(29,48,36,55),(30,47,37,54),(31,46,38,53),(32,45,39,52),(33,44,40,51),(34,43,41,50),(35,56,42,49)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 7A | ··· | 7F | 7G | ··· | 7O | 14A | ··· | 14F | 14G | ··· | 14O | 14P | ··· | 14U | 28A | ··· | 28F |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 7 | 7 | 7 | 7 | 49 | 49 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 14 | ··· | 14 | 14 | ··· | 14 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C4 | D7 | D7 | Dic7 | D14 | C4×D7 | D72 | D7×Dic7 |
| kernel | D7×Dic7 | C7×Dic7 | C7⋊Dic7 | D7×C14 | C7×D7 | Dic7 | D14 | D7 | C14 | C7 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 3 | 3 | 6 | 6 | 6 | 9 | 9 |
Matrix representation of D7×Dic7 ►in GL4(𝔽29) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 10 | 5 |
| 0 | 0 | 9 | 22 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 18 |
| 0 | 0 | 0 | 1 |
| 1 | 22 | 0 | 0 |
| 7 | 10 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 17 | 0 | 0 | 0 |
| 3 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,0,0,10,9,0,0,5,22],[28,0,0,0,0,28,0,0,0,0,28,0,0,0,18,1],[1,7,0,0,22,10,0,0,0,0,1,0,0,0,0,1],[17,3,0,0,0,12,0,0,0,0,1,0,0,0,0,1] >;
D7×Dic7 in GAP, Magma, Sage, TeX
D_7\times {\rm Dic}_7 % in TeX
G:=Group("D7xDic7"); // GroupNames label
G:=SmallGroup(392,18);
// by ID
G=gap.SmallGroup(392,18);
# by ID
G:=PCGroup([5,-2,-2,-2,-7,-7,26,488,8404]);
// Polycyclic
G:=Group<a,b,c,d|a^7=b^2=c^14=1,d^2=c^7,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
Export