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## G = D5×C14order 140 = 22·5·7

### Direct product of C14 and D5

Aliases: D5×C14, C10⋊C14, C703C2, C354C22, C5⋊(C2×C14), SmallGroup(140,9)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — D5×C14
 Chief series C1 — C5 — C35 — C7×D5 — D5×C14
 Lower central C5 — D5×C14
 Upper central C1 — 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

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