direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5×C14, C10⋊C14, C70⋊3C2, C35⋊4C22, C5⋊(C2×C14), SmallGroup(140,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C14 |
Generators and relations for D5×C14
G = < a,b,c | a14=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D5×C14
►On 70 points
Generators in S70
(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)(57 58 59 60 61 62 63 64 65 66 67 68 69 70)
(1 43 58 26 37)(2 44 59 27 38)(3 45 60 28 39)(4 46 61 15 40)(5 47 62 16 41)(6 48 63 17 42)(7 49 64 18 29)(8 50 65 19 30)(9 51 66 20 31)(10 52 67 21 32)(11 53 68 22 33)(12 54 69 23 34)(13 55 70 24 35)(14 56 57 25 36)
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 40)(12 41)(13 42)(14 29)(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)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)
G:=sub<Sym(70)| (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)(57,58,59,60,61,62,63,64,65,66,67,68,69,70), (1,43,58,26,37)(2,44,59,27,38)(3,45,60,28,39)(4,46,61,15,40)(5,47,62,16,41)(6,48,63,17,42)(7,49,64,18,29)(8,50,65,19,30)(9,51,66,20,31)(10,52,67,21,32)(11,53,68,22,33)(12,54,69,23,34)(13,55,70,24,35)(14,56,57,25,36), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,40)(12,41)(13,42)(14,29)(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)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)>;
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)(57,58,59,60,61,62,63,64,65,66,67,68,69,70), (1,43,58,26,37)(2,44,59,27,38)(3,45,60,28,39)(4,46,61,15,40)(5,47,62,16,41)(6,48,63,17,42)(7,49,64,18,29)(8,50,65,19,30)(9,51,66,20,31)(10,52,67,21,32)(11,53,68,22,33)(12,54,69,23,34)(13,55,70,24,35)(14,56,57,25,36), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,40)(12,41)(13,42)(14,29)(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)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70) );
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),(57,58,59,60,61,62,63,64,65,66,67,68,69,70)], [(1,43,58,26,37),(2,44,59,27,38),(3,45,60,28,39),(4,46,61,15,40),(5,47,62,16,41),(6,48,63,17,42),(7,49,64,18,29),(8,50,65,19,30),(9,51,66,20,31),(10,52,67,21,32),(11,53,68,22,33),(12,54,69,23,34),(13,55,70,24,35),(14,56,57,25,36)], [(1,30),(2,31),(3,32),(4,33),(5,34),(6,35),(7,36),(8,37),(9,38),(10,39),(11,40),(12,41),(13,42),(14,29),(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),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70)]])
D5×C14 is a maximal subgroup of
C35⋊D4 C7⋊D20
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 5A | 5B | 7A | ··· | 7F | 10A | 10B | 14A | ··· | 14F | 14G | ··· | 14R | 35A | ··· | 35L | 70A | ··· | 70L |
| order | 1 | 2 | 2 | 2 | 5 | 5 | 7 | ··· | 7 | 10 | 10 | 14 | ··· | 14 | 14 | ··· | 14 | 35 | ··· | 35 | 70 | ··· | 70 |
| size | 1 | 1 | 5 | 5 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | ··· | 2 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C7 | C14 | C14 | D5 | D10 | C7×D5 | D5×C14 |
| kernel | D5×C14 | C7×D5 | C70 | D10 | D5 | C10 | C14 | C7 | C2 | C1 |
| # reps | 1 | 2 | 1 | 6 | 12 | 6 | 2 | 2 | 12 | 12 |
Matrix representation of D5×C14 ►in GL2(𝔽29) generated by
| 5 | 0 |
| 0 | 5 |
| 6 | 22 |
| 1 | 28 |
| 28 | 0 |
| 28 | 1 |
G:=sub<GL(2,GF(29))| [5,0,0,5],[6,1,22,28],[28,28,0,1] >;
D5×C14 in GAP, Magma, Sage, TeX
D_5\times C_{14} % in TeX
G:=Group("D5xC14"); // GroupNames label
G:=SmallGroup(140,9);
// by ID
G=gap.SmallGroup(140,9);
# by ID
G:=PCGroup([4,-2,-2,-7,-5,1795]);
// Polycyclic
G:=Group<a,b,c|a^14=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export