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## G = C10×D7order 140 = 22·5·7

### Direct product of C10 and D7

Aliases: C10×D7, C14⋊C10, C702C2, C353C22, C7⋊(C2×C10), SmallGroup(140,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C10×D7
 Chief series C1 — C7 — C35 — C5×D7 — C10×D7
 Lower central C7 — C10×D7
 Upper central C1 — C10

Generators and relations for C10×D7
G = < a,b,c | a10=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C10×D7
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 43 18 31 30 55 62)(2 44 19 32 21 56 63)(3 45 20 33 22 57 64)(4 46 11 34 23 58 65)(5 47 12 35 24 59 66)(6 48 13 36 25 60 67)(7 49 14 37 26 51 68)(8 50 15 38 27 52 69)(9 41 16 39 28 53 70)(10 42 17 40 29 54 61)
(1 67)(2 68)(3 69)(4 70)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 28)(12 29)(13 30)(14 21)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(31 36)(32 37)(33 38)(34 39)(35 40)(41 58)(42 59)(43 60)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)(50 57)

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

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

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

C10×D7 is a maximal subgroup of   C35⋊D4  C5⋊D28

50 conjugacy classes

 class 1 2A 2B 2C 5A 5B 5C 5D 7A 7B 7C 10A 10B 10C 10D 10E ··· 10L 14A 14B 14C 35A ··· 35L 70A ··· 70L order 1 2 2 2 5 5 5 5 7 7 7 10 10 10 10 10 ··· 10 14 14 14 35 ··· 35 70 ··· 70 size 1 1 7 7 1 1 1 1 2 2 2 1 1 1 1 7 ··· 7 2 2 2 2 ··· 2 2 ··· 2

50 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C5 C10 C10 D7 D14 C5×D7 C10×D7 kernel C10×D7 C5×D7 C70 D14 D7 C14 C10 C5 C2 C1 # reps 1 2 1 4 8 4 3 3 12 12

Matrix representation of C10×D7 in GL2(𝔽41) generated by

 23 0 0 23
,
 10 37 25 27
,
 27 16 16 14
G:=sub<GL(2,GF(41))| [23,0,0,23],[10,25,37,27],[27,16,16,14] >;

C10×D7 in GAP, Magma, Sage, TeX

C_{10}\times D_7
% in TeX

G:=Group("C10xD7");
// GroupNames label

G:=SmallGroup(140,8);
// by ID

G=gap.SmallGroup(140,8);
# by ID

G:=PCGroup([4,-2,-2,-5,-7,1923]);
// Polycyclic

G:=Group<a,b,c|a^10=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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