Copied to
clipboard

## G = C5×D35order 350 = 2·52·7

### Direct product of C5 and D35

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

 Derived series C1 — C35 — C5×D35
 Chief series C1 — C7 — C35 — C5×C35 — C5×D35
 Lower central C35 — C5×D35
 Upper central C1 — C5

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

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

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

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

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

95 conjugacy classes

 class 1 2 5A 5B 5C 5D 5E ··· 5N 7A 7B 7C 10A 10B 10C 10D 35A ··· 35BT order 1 2 5 5 5 5 5 ··· 5 7 7 7 10 10 10 10 35 ··· 35 size 1 35 1 1 1 1 2 ··· 2 2 2 2 35 35 35 35 2 ··· 2

95 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 type + + + + + image C1 C2 C5 C10 D5 D7 C5×D5 C5×D7 D35 C5×D35 kernel C5×D35 C5×C35 D35 C35 C35 C52 C7 C5 C5 C1 # reps 1 1 4 4 2 3 8 12 12 48

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

 54 0 0 54
,
 60 0 65 58
,
 13 28 65 58
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

׿
×
𝔽