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G = C5×D35order 350 = 2·52·7

Direct product of C5 and D35

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

Aliases: C5×D35, C352D5, C351C10, C521D7, C7⋊(C5×D5), C5⋊(C5×D7), (C5×C35)⋊2C2, SmallGroup(350,7)

Series: Derived Chief Lower central Upper central

C1C35 — C5×D35
C1C7C35C5×C35 — C5×D35
C35 — C5×D35
C1C5

Generators and relations for C5×D35
 G = < a,b,c | a5=b35=c2=1, ab=ba, ac=ca, cbc=b-1 >

35C2
2C5
2C5
7D5
35C10
5D7
2C35
2C35
7C5×D5
5C5×D7

Smallest permutation representation of C5×D35
On 70 points
Generators in S70
(1 15 29 8 22)(2 16 30 9 23)(3 17 31 10 24)(4 18 32 11 25)(5 19 33 12 26)(6 20 34 13 27)(7 21 35 14 28)(36 57 43 64 50)(37 58 44 65 51)(38 59 45 66 52)(39 60 46 67 53)(40 61 47 68 54)(41 62 48 69 55)(42 63 49 70 56)
(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 51)(2 50)(3 49)(4 48)(5 47)(6 46)(7 45)(8 44)(9 43)(10 42)(11 41)(12 40)(13 39)(14 38)(15 37)(16 36)(17 70)(18 69)(19 68)(20 67)(21 66)(22 65)(23 64)(24 63)(25 62)(26 61)(27 60)(28 59)(29 58)(30 57)(31 56)(32 55)(33 54)(34 53)(35 52)

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

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

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

95 conjugacy classes

class 1  2 5A5B5C5D5E···5N7A7B7C10A10B10C10D35A···35BT
order1255555···57771010101035···35
size13511112···2222353535352···2

95 irreducible representations

dim1111222222
type+++++
imageC1C2C5C10D5D7C5×D5C5×D7D35C5×D35
kernelC5×D35C5×C35D35C35C35C52C7C5C5C1
# reps1144238121248

Matrix representation of C5×D35 in GL2(𝔽71) generated by

540
054
,
600
6558
,
1328
6558
G:=sub<GL(2,GF(71))| [54,0,0,54],[60,65,0,58],[13,65,28,58] >;

C5×D35 in GAP, Magma, Sage, TeX

C_5\times D_{35}
% in TeX

G:=Group("C5xD35");
// GroupNames label

G:=SmallGroup(350,7);
// by ID

G=gap.SmallGroup(350,7);
# by ID

G:=PCGroup([4,-2,-5,-5,-7,482,4803]);
// Polycyclic

G:=Group<a,b,c|a^5=b^35=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C5×D35 in TeX

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