direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C10×D7, C14⋊C10, C70⋊2C2, C35⋊3C22, C7⋊(C2×C10), SmallGroup(140,8)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C10×D7 |
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 60 31 23 62 13)(2 44 51 32 24 63 14)(3 45 52 33 25 64 15)(4 46 53 34 26 65 16)(5 47 54 35 27 66 17)(6 48 55 36 28 67 18)(7 49 56 37 29 68 19)(8 50 57 38 30 69 20)(9 41 58 39 21 70 11)(10 42 59 40 22 61 12)
(1 18)(2 19)(3 20)(4 11)(5 12)(6 13)(7 14)(8 15)(9 16)(10 17)(21 53)(22 54)(23 55)(24 56)(25 57)(26 58)(27 59)(28 60)(29 51)(30 52)(31 36)(32 37)(33 38)(34 39)(35 40)(41 65)(42 66)(43 67)(44 68)(45 69)(46 70)(47 61)(48 62)(49 63)(50 64)
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), (1,43,60,31,23,62,13)(2,44,51,32,24,63,14)(3,45,52,33,25,64,15)(4,46,53,34,26,65,16)(5,47,54,35,27,66,17)(6,48,55,36,28,67,18)(7,49,56,37,29,68,19)(8,50,57,38,30,69,20)(9,41,58,39,21,70,11)(10,42,59,40,22,61,12), (1,18)(2,19)(3,20)(4,11)(5,12)(6,13)(7,14)(8,15)(9,16)(10,17)(21,53)(22,54)(23,55)(24,56)(25,57)(26,58)(27,59)(28,60)(29,51)(30,52)(31,36)(32,37)(33,38)(34,39)(35,40)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,61)(48,62)(49,63)(50,64)>;
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), (1,43,60,31,23,62,13)(2,44,51,32,24,63,14)(3,45,52,33,25,64,15)(4,46,53,34,26,65,16)(5,47,54,35,27,66,17)(6,48,55,36,28,67,18)(7,49,56,37,29,68,19)(8,50,57,38,30,69,20)(9,41,58,39,21,70,11)(10,42,59,40,22,61,12), (1,18)(2,19)(3,20)(4,11)(5,12)(6,13)(7,14)(8,15)(9,16)(10,17)(21,53)(22,54)(23,55)(24,56)(25,57)(26,58)(27,59)(28,60)(29,51)(30,52)(31,36)(32,37)(33,38)(34,39)(35,40)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,61)(48,62)(49,63)(50,64) );
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)], [(1,43,60,31,23,62,13),(2,44,51,32,24,63,14),(3,45,52,33,25,64,15),(4,46,53,34,26,65,16),(5,47,54,35,27,66,17),(6,48,55,36,28,67,18),(7,49,56,37,29,68,19),(8,50,57,38,30,69,20),(9,41,58,39,21,70,11),(10,42,59,40,22,61,12)], [(1,18),(2,19),(3,20),(4,11),(5,12),(6,13),(7,14),(8,15),(9,16),(10,17),(21,53),(22,54),(23,55),(24,56),(25,57),(26,58),(27,59),(28,60),(29,51),(30,52),(31,36),(32,37),(33,38),(34,39),(35,40),(41,65),(42,66),(43,67),(44,68),(45,69),(46,70),(47,61),(48,62),(49,63),(50,64)]])
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
Export