Copied to
clipboard

G = C7×D35order 490 = 2·5·72

Direct product of C7 and D35

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

Aliases: C7×D35, C353D7, C351C14, C721D5, C5⋊(C7×D7), C7⋊(C7×D5), (C7×C35)⋊2C2, SmallGroup(490,8)

Series: Derived Chief Lower central Upper central

C1C35 — C7×D35
C1C7C35C7×C35 — C7×D35
C35 — C7×D35
C1C7

Generators and relations for C7×D35
 G = < a,b,c | a7=b35=c2=1, ab=ba, ac=ca, cbc=b-1 >

35C2
2C7
2C7
2C7
7D5
5D7
35C14
2C35
2C35
2C35
7C7×D5
5C7×D7

Smallest permutation representation of C7×D35
On 70 points
Generators in S70
(1 31 26 21 16 11 6)(2 32 27 22 17 12 7)(3 33 28 23 18 13 8)(4 34 29 24 19 14 9)(5 35 30 25 20 15 10)(36 41 46 51 56 61 66)(37 42 47 52 57 62 67)(38 43 48 53 58 63 68)(39 44 49 54 59 64 69)(40 45 50 55 60 65 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 60)(2 59)(3 58)(4 57)(5 56)(6 55)(7 54)(8 53)(9 52)(10 51)(11 50)(12 49)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 42)(20 41)(21 40)(22 39)(23 38)(24 37)(25 36)(26 70)(27 69)(28 68)(29 67)(30 66)(31 65)(32 64)(33 63)(34 62)(35 61)

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

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

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

133 conjugacy classes

class 1  2 5A5B7A···7F7G···7AA14A···14F35A···35CR
order12557···77···714···1435···35
size135221···12···235···352···2

133 irreducible representations

dim1111222222
type+++++
imageC1C2C7C14D5D7C7×D5D35C7×D7C7×D35
kernelC7×D35C7×C35D35C35C72C35C7C7C5C1
# reps11662312121872

Matrix representation of C7×D35 in GL2(𝔽71) generated by

200
020
,
100
5664
,
967
2062
G:=sub<GL(2,GF(71))| [20,0,0,20],[10,56,0,64],[9,20,67,62] >;

C7×D35 in GAP, Magma, Sage, TeX

C_7\times D_{35}
% in TeX

G:=Group("C7xD35");
// GroupNames label

G:=SmallGroup(490,8);
// by ID

G=gap.SmallGroup(490,8);
# by ID

G:=PCGroup([4,-2,-7,-5,-7,674,6723]);
// Polycyclic

G:=Group<a,b,c|a^7=b^35=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C7×D35 in TeX

׿
×
𝔽