direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C10×F7, C14⋊C30, D7⋊C30, D14⋊C15, C70⋊2C6, C7⋊(C2×C30), (C10×D7)⋊C3, C35⋊3(C2×C6), (C5×D7)⋊2C6, C7⋊C3⋊(C2×C10), (C2×C7⋊C3)⋊C10, (C10×C7⋊C3)⋊2C2, (C5×C7⋊C3)⋊3C22, SmallGroup(420,17)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C10×F7 |
Generators and relations for C10×F7
G = < a,b,c | a10=b7=c6=1, ab=ba, ac=ca, cbc-1=b5 >
Smallest permutation representation of C10×F7
►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 34 13 65 52 22 43)(2 35 14 66 53 23 44)(3 36 15 67 54 24 45)(4 37 16 68 55 25 46)(5 38 17 69 56 26 47)(6 39 18 70 57 27 48)(7 40 19 61 58 28 49)(8 31 20 62 59 29 50)(9 32 11 63 60 30 41)(10 33 12 64 51 21 42)
(11 41 60 30 32 63)(12 42 51 21 33 64)(13 43 52 22 34 65)(14 44 53 23 35 66)(15 45 54 24 36 67)(16 46 55 25 37 68)(17 47 56 26 38 69)(18 48 57 27 39 70)(19 49 58 28 40 61)(20 50 59 29 31 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), (1,34,13,65,52,22,43)(2,35,14,66,53,23,44)(3,36,15,67,54,24,45)(4,37,16,68,55,25,46)(5,38,17,69,56,26,47)(6,39,18,70,57,27,48)(7,40,19,61,58,28,49)(8,31,20,62,59,29,50)(9,32,11,63,60,30,41)(10,33,12,64,51,21,42), (11,41,60,30,32,63)(12,42,51,21,33,64)(13,43,52,22,34,65)(14,44,53,23,35,66)(15,45,54,24,36,67)(16,46,55,25,37,68)(17,47,56,26,38,69)(18,48,57,27,39,70)(19,49,58,28,40,61)(20,50,59,29,31,62)>;
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), (1,34,13,65,52,22,43)(2,35,14,66,53,23,44)(3,36,15,67,54,24,45)(4,37,16,68,55,25,46)(5,38,17,69,56,26,47)(6,39,18,70,57,27,48)(7,40,19,61,58,28,49)(8,31,20,62,59,29,50)(9,32,11,63,60,30,41)(10,33,12,64,51,21,42), (11,41,60,30,32,63)(12,42,51,21,33,64)(13,43,52,22,34,65)(14,44,53,23,35,66)(15,45,54,24,36,67)(16,46,55,25,37,68)(17,47,56,26,38,69)(18,48,57,27,39,70)(19,49,58,28,40,61)(20,50,59,29,31,62) );
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)], [(1,34,13,65,52,22,43),(2,35,14,66,53,23,44),(3,36,15,67,54,24,45),(4,37,16,68,55,25,46),(5,38,17,69,56,26,47),(6,39,18,70,57,27,48),(7,40,19,61,58,28,49),(8,31,20,62,59,29,50),(9,32,11,63,60,30,41),(10,33,12,64,51,21,42)], [(11,41,60,30,32,63),(12,42,51,21,33,64),(13,43,52,22,34,65),(14,44,53,23,35,66),(15,45,54,24,36,67),(16,46,55,25,37,68),(17,47,56,26,38,69),(18,48,57,27,39,70),(19,49,58,28,40,61),(20,50,59,29,31,62)]])
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 5A | 5B | 5C | 5D | 6A | ··· | 6F | 7 | 10A | 10B | 10C | 10D | 10E | ··· | 10L | 14 | 15A | ··· | 15H | 30A | ··· | 30X | 35A | 35B | 35C | 35D | 70A | 70B | 70C | 70D |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 7 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 14 | 15 | ··· | 15 | 30 | ··· | 30 | 35 | 35 | 35 | 35 | 70 | 70 | 70 | 70 |
| size | 1 | 1 | 7 | 7 | 7 | 7 | 1 | 1 | 1 | 1 | 7 | ··· | 7 | 6 | 1 | 1 | 1 | 1 | 7 | ··· | 7 | 6 | 7 | ··· | 7 | 7 | ··· | 7 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C3 | C5 | C6 | C6 | C10 | C10 | C15 | C30 | C30 | F7 | C2×F7 | C5×F7 | C10×F7 |
| kernel | C10×F7 | C5×F7 | C10×C7⋊C3 | C10×D7 | C2×F7 | C5×D7 | C70 | F7 | C2×C7⋊C3 | D14 | D7 | C14 | C10 | C5 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 4 | 8 | 16 | 8 | 1 | 1 | 4 | 4 |
Matrix representation of C10×F7 ►in GL7(𝔽211)
| 23 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 210 |
| 0 | 1 | 0 | 0 | 0 | 0 | 210 |
| 0 | 0 | 1 | 0 | 0 | 0 | 210 |
| 0 | 0 | 0 | 1 | 0 | 0 | 210 |
| 0 | 0 | 0 | 0 | 1 | 0 | 210 |
| 0 | 0 | 0 | 0 | 0 | 1 | 210 |
| 197 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
G:=sub<GL(7,GF(211))| [23,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,210,210,210,210,210,210],[197,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0] >;
C10×F7 in GAP, Magma, Sage, TeX
C_{10}\times F_7 % in TeX
G:=Group("C10xF7"); // GroupNames label
G:=SmallGroup(420,17);
// by ID
G=gap.SmallGroup(420,17);
# by ID
G:=PCGroup([5,-2,-2,-3,-5,-7,9004,1514]);
// Polycyclic
G:=Group<a,b,c|a^10=b^7=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations
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