direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C7⋊F5, C14⋊F5, C70⋊1C4, D5⋊Dic7, D10.D7, C10⋊Dic7, D5.2D14, C5⋊(C2×Dic7), C7⋊2(C2×F5), C35⋊2(C2×C4), (C7×D5)⋊2C4, (D5×C14).2C2, (C7×D5).2C22, SmallGroup(280,35)
Series: Derived ►Chief ►Lower central ►Upper central
| C35 — C2×C7⋊F5 |
Generators and relations for C2×C7⋊F5
G = < a,b,c,d | a2=b7=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >
Smallest permutation representation of C2×C7⋊F5
►On 70 points
Generators in S70
(1 36)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(22 57)(23 58)(24 59)(25 60)(26 61)(27 62)(28 63)(29 64)(30 65)(31 66)(32 67)(33 68)(34 69)(35 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 29 22 15 8)(2 30 23 16 9)(3 31 24 17 10)(4 32 25 18 11)(5 33 26 19 12)(6 34 27 20 13)(7 35 28 21 14)(36 64 57 50 43)(37 65 58 51 44)(38 66 59 52 45)(39 67 60 53 46)(40 68 61 54 47)(41 69 62 55 48)(42 70 63 56 49)
(1 36)(2 42)(3 41)(4 40)(5 39)(6 38)(7 37)(8 50 29 57)(9 56 30 63)(10 55 31 62)(11 54 32 61)(12 53 33 60)(13 52 34 59)(14 51 35 58)(15 64 22 43)(16 70 23 49)(17 69 24 48)(18 68 25 47)(19 67 26 46)(20 66 27 45)(21 65 28 44)
G:=sub<Sym(70)| (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,63)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70), (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), (1,29,22,15,8)(2,30,23,16,9)(3,31,24,17,10)(4,32,25,18,11)(5,33,26,19,12)(6,34,27,20,13)(7,35,28,21,14)(36,64,57,50,43)(37,65,58,51,44)(38,66,59,52,45)(39,67,60,53,46)(40,68,61,54,47)(41,69,62,55,48)(42,70,63,56,49), (1,36)(2,42)(3,41)(4,40)(5,39)(6,38)(7,37)(8,50,29,57)(9,56,30,63)(10,55,31,62)(11,54,32,61)(12,53,33,60)(13,52,34,59)(14,51,35,58)(15,64,22,43)(16,70,23,49)(17,69,24,48)(18,68,25,47)(19,67,26,46)(20,66,27,45)(21,65,28,44)>;
G:=Group( (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,63)(29,64)(30,65)(31,66)(32,67)(33,68)(34,69)(35,70), (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), (1,29,22,15,8)(2,30,23,16,9)(3,31,24,17,10)(4,32,25,18,11)(5,33,26,19,12)(6,34,27,20,13)(7,35,28,21,14)(36,64,57,50,43)(37,65,58,51,44)(38,66,59,52,45)(39,67,60,53,46)(40,68,61,54,47)(41,69,62,55,48)(42,70,63,56,49), (1,36)(2,42)(3,41)(4,40)(5,39)(6,38)(7,37)(8,50,29,57)(9,56,30,63)(10,55,31,62)(11,54,32,61)(12,53,33,60)(13,52,34,59)(14,51,35,58)(15,64,22,43)(16,70,23,49)(17,69,24,48)(18,68,25,47)(19,67,26,46)(20,66,27,45)(21,65,28,44) );
G=PermutationGroup([[(1,36),(2,37),(3,38),(4,39),(5,40),(6,41),(7,42),(8,43),(9,44),(10,45),(11,46),(12,47),(13,48),(14,49),(15,50),(16,51),(17,52),(18,53),(19,54),(20,55),(21,56),(22,57),(23,58),(24,59),(25,60),(26,61),(27,62),(28,63),(29,64),(30,65),(31,66),(32,67),(33,68),(34,69),(35,70)], [(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)], [(1,29,22,15,8),(2,30,23,16,9),(3,31,24,17,10),(4,32,25,18,11),(5,33,26,19,12),(6,34,27,20,13),(7,35,28,21,14),(36,64,57,50,43),(37,65,58,51,44),(38,66,59,52,45),(39,67,60,53,46),(40,68,61,54,47),(41,69,62,55,48),(42,70,63,56,49)], [(1,36),(2,42),(3,41),(4,40),(5,39),(6,38),(7,37),(8,50,29,57),(9,56,30,63),(10,55,31,62),(11,54,32,61),(12,53,33,60),(13,52,34,59),(14,51,35,58),(15,64,22,43),(16,70,23,49),(17,69,24,48),(18,68,25,47),(19,67,26,46),(20,66,27,45),(21,65,28,44)]])
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5 | 7A | 7B | 7C | 10 | 14A | 14B | 14C | 14D | ··· | 14I | 35A | ··· | 35F | 70A | ··· | 70F |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 7 | 7 | 7 | 10 | 14 | 14 | 14 | 14 | ··· | 14 | 35 | ··· | 35 | 70 | ··· | 70 |
| size | 1 | 1 | 5 | 5 | 35 | 35 | 35 | 35 | 4 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | - | + | + | ||||
| image | C1 | C2 | C2 | C4 | C4 | D7 | Dic7 | D14 | Dic7 | F5 | C2×F5 | C7⋊F5 | C2×C7⋊F5 |
| kernel | C2×C7⋊F5 | C7⋊F5 | D5×C14 | C7×D5 | C70 | D10 | D5 | D5 | C10 | C14 | C7 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 1 | 1 | 6 | 6 |
Matrix representation of C2×C7⋊F5 ►in GL6(𝔽281)
| 280 | 0 | 0 | 0 | 0 | 0 |
| 0 | 280 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 274 | 280 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 47 | 280 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 47 | 280 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 280 | 0 | 14 | 107 |
| 0 | 0 | 0 | 280 | 174 | 266 |
| 53 | 0 | 0 | 0 | 0 | 0 |
| 191 | 228 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 47 | 280 | 0 | 0 |
| 0 | 0 | 267 | 174 | 267 | 174 |
| 0 | 0 | 78 | 14 | 78 | 14 |
G:=sub<GL(6,GF(281))| [280,0,0,0,0,0,0,280,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[274,1,0,0,0,0,280,0,0,0,0,0,0,0,47,1,0,0,0,0,280,0,0,0,0,0,0,0,47,1,0,0,0,0,280,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,280,0,0,0,0,0,0,280,0,0,1,0,14,174,0,0,0,1,107,266],[53,191,0,0,0,0,0,228,0,0,0,0,0,0,1,47,267,78,0,0,0,280,174,14,0,0,0,0,267,78,0,0,0,0,174,14] >;
C2×C7⋊F5 in GAP, Magma, Sage, TeX
C_2\times C_7\rtimes F_5
% in TeX
G:=Group("C2xC7:F5"); // GroupNames label
G:=SmallGroup(280,35);
// by ID
G=gap.SmallGroup(280,35);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-7,20,483,173,6004]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^7=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations
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