direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C10×C7⋊C3, C70⋊C3, C14⋊C15, C35⋊4C6, C7⋊2C30, SmallGroup(210,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C10×C7⋊C3 |
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
C35⋊3C12
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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