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G = D5×C14order 140 = 22·5·7

Direct product of C14 and D5

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

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

Series: Derived Chief Lower central Upper central

C1C5 — D5×C14
C1C5C35C7×D5 — D5×C14
C5 — D5×C14
C1C14

Generators and relations for D5×C14
 G = < a,b,c | a14=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

5C2
5C2
5C22
5C14
5C14
5C2×C14

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 18 51 59 33)(2 19 52 60 34)(3 20 53 61 35)(4 21 54 62 36)(5 22 55 63 37)(6 23 56 64 38)(7 24 43 65 39)(8 25 44 66 40)(9 26 45 67 41)(10 27 46 68 42)(11 28 47 69 29)(12 15 48 70 30)(13 16 49 57 31)(14 17 50 58 32)
(1 40)(2 41)(3 42)(4 29)(5 30)(6 31)(7 32)(8 33)(9 34)(10 35)(11 36)(12 37)(13 38)(14 39)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 57)(24 58)(25 59)(26 60)(27 61)(28 62)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)

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), (1,18,51,59,33)(2,19,52,60,34)(3,20,53,61,35)(4,21,54,62,36)(5,22,55,63,37)(6,23,56,64,38)(7,24,43,65,39)(8,25,44,66,40)(9,26,45,67,41)(10,27,46,68,42)(11,28,47,69,29)(12,15,48,70,30)(13,16,49,57,31)(14,17,50,58,32), (1,40)(2,41)(3,42)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,57)(24,58)(25,59)(26,60)(27,61)(28,62)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)>;

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), (1,18,51,59,33)(2,19,52,60,34)(3,20,53,61,35)(4,21,54,62,36)(5,22,55,63,37)(6,23,56,64,38)(7,24,43,65,39)(8,25,44,66,40)(9,26,45,67,41)(10,27,46,68,42)(11,28,47,69,29)(12,15,48,70,30)(13,16,49,57,31)(14,17,50,58,32), (1,40)(2,41)(3,42)(4,29)(5,30)(6,31)(7,32)(8,33)(9,34)(10,35)(11,36)(12,37)(13,38)(14,39)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,57)(24,58)(25,59)(26,60)(27,61)(28,62)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56) );

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)], [(1,18,51,59,33),(2,19,52,60,34),(3,20,53,61,35),(4,21,54,62,36),(5,22,55,63,37),(6,23,56,64,38),(7,24,43,65,39),(8,25,44,66,40),(9,26,45,67,41),(10,27,46,68,42),(11,28,47,69,29),(12,15,48,70,30),(13,16,49,57,31),(14,17,50,58,32)], [(1,40),(2,41),(3,42),(4,29),(5,30),(6,31),(7,32),(8,33),(9,34),(10,35),(11,36),(12,37),(13,38),(14,39),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,57),(24,58),(25,59),(26,60),(27,61),(28,62),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(49,56)])

D5×C14 is a maximal subgroup of   C35⋊D4  C7⋊D20

56 conjugacy classes

class 1 2A2B2C5A5B7A···7F10A10B14A···14F14G···14R35A···35L70A···70L
order1222557···7101014···1414···1435···3570···70
size1155221···1221···15···52···22···2

56 irreducible representations

dim1111112222
type+++++
imageC1C2C2C7C14C14D5D10C7×D5D5×C14
kernelD5×C14C7×D5C70D10D5C10C14C7C2C1
# reps1216126221212

Matrix representation of D5×C14 in GL2(𝔽29) generated by

50
05
,
622
128
,
280
281
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

Subgroup lattice of D5×C14 in TeX

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