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G = D5×C35order 350 = 2·52·7

Direct product of C35 and D5

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — D5×C35
 Chief series C1 — C5 — C35 — C5×C35 — D5×C35
 Lower central C5 — D5×C35
 Upper central C1 — C35

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

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

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

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

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

140 conjugacy classes

 class 1 2 5A 5B 5C 5D 5E ··· 5N 7A ··· 7F 10A 10B 10C 10D 14A ··· 14F 35A ··· 35X 35Y ··· 35CF 70A ··· 70X order 1 2 5 5 5 5 5 ··· 5 7 ··· 7 10 10 10 10 14 ··· 14 35 ··· 35 35 ··· 35 70 ··· 70 size 1 5 1 1 1 1 2 ··· 2 1 ··· 1 5 5 5 5 5 ··· 5 1 ··· 1 2 ··· 2 5 ··· 5

140 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 type + + + image C1 C2 C5 C7 C10 C14 C35 C70 D5 C5×D5 C7×D5 D5×C35 kernel D5×C35 C5×C35 C7×D5 C5×D5 C35 C52 D5 C5 C35 C7 C5 C1 # reps 1 1 4 6 4 6 24 24 2 8 12 48

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

 12 0 0 12
,
 25 0 55 54
,
 17 18 55 54
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

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