direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C2×C5⋊F7, D70⋊C3, C10⋊F7, C70⋊1C6, D35⋊2C6, C14⋊(C3×D5), C7⋊C3⋊2D10, C7⋊2(C6×D5), C5⋊2(C2×F7), C35⋊2(C2×C6), (C2×C7⋊C3)⋊D5, (C10×C7⋊C3)⋊1C2, (C5×C7⋊C3)⋊2C22, SmallGroup(420,19)
Series: Derived ►Chief ►Lower central ►Upper central
| C35 — C2×C5⋊F7 |
Generators and relations for C2×C5⋊F7
G = < a,b,c,d | a2=b5=c7=d6=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c5 >
Smallest permutation representation of C2×C5⋊F7
►On 70 points
Generators in S70
(1 41)(2 42)(3 36)(4 37)(5 38)(6 39)(7 40)(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 34 27 20 13)(2 35 28 21 14)(3 29 22 15 8)(4 30 23 16 9)(5 31 24 17 10)(6 32 25 18 11)(7 33 26 19 12)(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 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)
(2 4 3 7 5 6)(8 33 10 32 14 30)(9 29 12 31 11 35)(13 34)(15 26 17 25 21 23)(16 22 19 24 18 28)(20 27)(36 40 38 39 42 37)(43 68 45 67 49 65)(44 64 47 66 46 70)(48 69)(50 61 52 60 56 58)(51 57 54 59 53 63)(55 62)
G:=sub<Sym(70)| (1,41)(2,42)(3,36)(4,37)(5,38)(6,39)(7,40)(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), (1,34,27,20,13)(2,35,28,21,14)(3,29,22,15,8)(4,30,23,16,9)(5,31,24,17,10)(6,32,25,18,11)(7,33,26,19,12)(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), (1,2,3,4,5,6,7)(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), (2,4,3,7,5,6)(8,33,10,32,14,30)(9,29,12,31,11,35)(13,34)(15,26,17,25,21,23)(16,22,19,24,18,28)(20,27)(36,40,38,39,42,37)(43,68,45,67,49,65)(44,64,47,66,46,70)(48,69)(50,61,52,60,56,58)(51,57,54,59,53,63)(55,62)>;
G:=Group( (1,41)(2,42)(3,36)(4,37)(5,38)(6,39)(7,40)(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), (1,34,27,20,13)(2,35,28,21,14)(3,29,22,15,8)(4,30,23,16,9)(5,31,24,17,10)(6,32,25,18,11)(7,33,26,19,12)(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), (1,2,3,4,5,6,7)(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), (2,4,3,7,5,6)(8,33,10,32,14,30)(9,29,12,31,11,35)(13,34)(15,26,17,25,21,23)(16,22,19,24,18,28)(20,27)(36,40,38,39,42,37)(43,68,45,67,49,65)(44,64,47,66,46,70)(48,69)(50,61,52,60,56,58)(51,57,54,59,53,63)(55,62) );
G=PermutationGroup([[(1,41),(2,42),(3,36),(4,37),(5,38),(6,39),(7,40),(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)], [(1,34,27,20,13),(2,35,28,21,14),(3,29,22,15,8),(4,30,23,16,9),(5,31,24,17,10),(6,32,25,18,11),(7,33,26,19,12),(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)], [(1,2,3,4,5,6,7),(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)], [(2,4,3,7,5,6),(8,33,10,32,14,30),(9,29,12,31,11,35),(13,34),(15,26,17,25,21,23),(16,22,19,24,18,28),(20,27),(36,40,38,39,42,37),(43,68,45,67,49,65),(44,64,47,66,46,70),(48,69),(50,61,52,60,56,58),(51,57,54,59,53,63),(55,62)]])
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 7 | 10A | 10B | 14 | 15A | 15B | 15C | 15D | 30A | 30B | 30C | 30D | 35A | 35B | 35C | 35D | 70A | 70B | 70C | 70D |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 7 | 10 | 10 | 14 | 15 | 15 | 15 | 15 | 30 | 30 | 30 | 30 | 35 | 35 | 35 | 35 | 70 | 70 | 70 | 70 |
| size | 1 | 1 | 35 | 35 | 7 | 7 | 2 | 2 | 7 | 7 | 35 | 35 | 35 | 35 | 6 | 2 | 2 | 6 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C3 | C6 | C6 | D5 | D10 | C3×D5 | C6×D5 | F7 | C2×F7 | C5⋊F7 | C2×C5⋊F7 |
| kernel | C2×C5⋊F7 | C5⋊F7 | C10×C7⋊C3 | D70 | D35 | C70 | C2×C7⋊C3 | C7⋊C3 | C14 | C7 | C10 | C5 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 4 | 4 |
Matrix representation of C2×C5⋊F7 ►in GL6(𝔽211)
| 210 | 0 | 0 | 0 | 0 | 0 |
| 0 | 210 | 0 | 0 | 0 | 0 |
| 0 | 0 | 210 | 0 | 0 | 0 |
| 0 | 0 | 0 | 210 | 0 | 0 |
| 0 | 0 | 0 | 0 | 210 | 0 |
| 0 | 0 | 0 | 0 | 0 | 210 |
| 74 | 181 | 181 | 0 | 181 | 0 |
| 0 | 74 | 181 | 181 | 0 | 181 |
| 30 | 30 | 104 | 0 | 0 | 30 |
| 181 | 0 | 0 | 74 | 181 | 181 |
| 30 | 0 | 30 | 30 | 104 | 0 |
| 0 | 30 | 0 | 30 | 30 | 104 |
| 0 | 1 | 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 |
| 210 | 210 | 210 | 210 | 210 | 210 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 210 | 210 | 210 | 210 | 210 | 210 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(211))| [210,0,0,0,0,0,0,210,0,0,0,0,0,0,210,0,0,0,0,0,0,210,0,0,0,0,0,0,210,0,0,0,0,0,0,210],[74,0,30,181,30,0,181,74,30,0,0,30,181,181,104,0,30,0,0,181,0,74,30,30,181,0,0,181,104,30,0,181,30,181,0,104],[0,0,0,0,0,210,1,0,0,0,0,210,0,1,0,0,0,210,0,0,1,0,0,210,0,0,0,1,0,210,0,0,0,0,1,210],[1,0,0,0,210,0,0,0,0,1,210,0,0,0,0,0,210,0,0,0,1,0,210,0,0,0,0,0,210,1,0,1,0,0,210,0] >;
C2×C5⋊F7 in GAP, Magma, Sage, TeX
C_2\times C_5\rtimes F_7
% in TeX
G:=Group("C2xC5:F7"); // GroupNames label
G:=SmallGroup(420,19);
// by ID
G=gap.SmallGroup(420,19);
# by ID
G:=PCGroup([5,-2,-2,-3,-5,-7,963,9004,764]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^5=c^7=d^6=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^5>;
// generators/relations
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