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## G = C10×C7⋊C3order 210 = 2·3·5·7

### Direct product of C10 and C7⋊C3

Aliases: C10×C7⋊C3, C70⋊C3, C14⋊C15, C354C6, C72C30, SmallGroup(210,4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C10×C7⋊C3
 Chief series C1 — C7 — C35 — C5×C7⋊C3 — C10×C7⋊C3
 Lower central C7 — C10×C7⋊C3
 Upper central C1 — C10

Generators and relations for C10×C7⋊C3
G = < a,b,c | a10=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >

Smallest permutation representation of C10×C7⋊C3
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 13 31 58 69 22 43)(2 14 32 59 70 23 44)(3 15 33 60 61 24 45)(4 16 34 51 62 25 46)(5 17 35 52 63 26 47)(6 18 36 53 64 27 48)(7 19 37 54 65 28 49)(8 20 38 55 66 29 50)(9 11 39 56 67 30 41)(10 12 40 57 68 21 42)
(11 39 67)(12 40 68)(13 31 69)(14 32 70)(15 33 61)(16 34 62)(17 35 63)(18 36 64)(19 37 65)(20 38 66)(21 57 42)(22 58 43)(23 59 44)(24 60 45)(25 51 46)(26 52 47)(27 53 48)(28 54 49)(29 55 50)(30 56 41)

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

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

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

C10×C7⋊C3 is a maximal subgroup of   C353C12

50 conjugacy classes

 class 1 2 3A 3B 5A 5B 5C 5D 6A 6B 7A 7B 10A 10B 10C 10D 14A 14B 15A ··· 15H 30A ··· 30H 35A ··· 35H 70A ··· 70H order 1 2 3 3 5 5 5 5 6 6 7 7 10 10 10 10 14 14 15 ··· 15 30 ··· 30 35 ··· 35 70 ··· 70 size 1 1 7 7 1 1 1 1 7 7 3 3 1 1 1 1 3 3 7 ··· 7 7 ··· 7 3 ··· 3 3 ··· 3

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 3 3 type + + image C1 C2 C3 C5 C6 C10 C15 C30 C7⋊C3 C2×C7⋊C3 C5×C7⋊C3 C10×C7⋊C3 kernel C10×C7⋊C3 C5×C7⋊C3 C70 C2×C7⋊C3 C35 C7⋊C3 C14 C7 C10 C5 C2 C1 # reps 1 1 2 4 2 4 8 8 2 2 8 8

Matrix representation of C10×C7⋊C3 in GL3(𝔽11) generated by

 7 0 0 0 7 0 0 0 7
,
 4 0 10 9 0 0 6 6 0
,
 1 8 0 0 10 2 0 5 0
G:=sub<GL(3,GF(11))| [7,0,0,0,7,0,0,0,7],[4,9,6,0,0,6,10,0,0],[1,0,0,8,10,5,0,2,0] >;

C10×C7⋊C3 in GAP, Magma, Sage, TeX

C_{10}\times C_7\rtimes C_3
% in TeX

G:=Group("C10xC7:C3");
// GroupNames label

G:=SmallGroup(210,4);
// by ID

G=gap.SmallGroup(210,4);
# by ID

G:=PCGroup([4,-2,-3,-5,-7,487]);
// Polycyclic

G:=Group<a,b,c|a^10=b^7=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

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