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G = D5×C35order 350 = 2·52·7

Direct product of C35 and D5

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

Aliases: D5×C35, C5⋊C70, C353C10, C521C14, (C5×C35)⋊4C2, SmallGroup(350,6)

Series: Derived Chief Lower central Upper central

C1C5 — D5×C35
C1C5C35C5×C35 — D5×C35
C5 — D5×C35
C1C35

Generators and relations for D5×C35
 G = < a,b,c | a35=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

5C2
2C5
2C5
5C10
5C14
2C35
2C35
5C70

Smallest permutation representation of D5×C35
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 8 15 22 29)(2 9 16 23 30)(3 10 17 24 31)(4 11 18 25 32)(5 12 19 26 33)(6 13 20 27 34)(7 14 21 28 35)(36 64 57 50 43)(37 65 58 51 44)(38 66 59 52 45)(39 67 60 53 46)(40 68 61 54 47)(41 69 62 55 48)(42 70 63 56 49)
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 61)(12 62)(13 63)(14 64)(15 65)(16 66)(17 67)(18 68)(19 69)(20 70)(21 36)(22 37)(23 38)(24 39)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 49)(35 50)

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

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

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

140 conjugacy classes

class 1  2 5A5B5C5D5E···5N7A···7F10A10B10C10D14A···14F35A···35X35Y···35CF70A···70X
order1255555···57···71010101014···1435···3535···3570···70
size1511112···21···155555···51···12···25···5

140 irreducible representations

dim111111112222
type+++
imageC1C2C5C7C10C14C35C70D5C5×D5C7×D5D5×C35
kernelD5×C35C5×C35C7×D5C5×D5C35C52D5C5C35C7C5C1
# reps1146462424281248

Matrix representation of D5×C35 in GL2(𝔽71) generated by

120
012
,
250
5554
,
1718
5554
G:=sub<GL(2,GF(71))| [12,0,0,12],[25,55,0,54],[17,55,18,54] >;

D5×C35 in GAP, Magma, Sage, TeX

D_5\times C_{35}
% in TeX

G:=Group("D5xC35");
// GroupNames label

G:=SmallGroup(350,6);
// by ID

G=gap.SmallGroup(350,6);
# by ID

G:=PCGroup([4,-2,-5,-7,-5,4483]);
// Polycyclic

G:=Group<a,b,c|a^35=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×C35 in TeX

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