direct product, metacyclic, supersoluble, monomial, A-group
Aliases: C10×C5⋊C8, C10⋊C40, C102.1C4, Dic5.2C20, C5⋊2(C2×C40), (C5×C10)⋊2C8, C52⋊11(C2×C8), C2.3(C10×F5), (C2×C10).1C20, C10.5(C2×C20), (C2×C10).12F5, C10.46(C2×F5), C22.2(C5×F5), (C5×Dic5).5C4, Dic5.7(C2×C10), (C10×Dic5).3C2, (C2×Dic5).4C10, (C5×Dic5).12C22, (C5×C10).17(C2×C4), SmallGroup(400,139)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C10×C5⋊C8 |
Generators and relations for C10×C5⋊C8
G = < a,b,c | a10=b5=c8=1, ab=ba, ac=ca, cbc-1=b3 >
Smallest permutation representation of C10×C5⋊C8
►On 80 points
Generators in S80
(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)(71 72 73 74 75 76 77 78 79 80)
(1 3 5 7 9)(2 4 6 8 10)(11 19 17 15 13)(12 20 18 16 14)(21 27 23 29 25)(22 28 24 30 26)(31 33 35 37 39)(32 34 36 38 40)(41 49 47 45 43)(42 50 48 46 44)(51 55 59 53 57)(52 56 60 54 58)(61 67 63 69 65)(62 68 64 70 66)(71 75 79 73 77)(72 76 80 74 78)
(1 26 45 72 37 65 13 58)(2 27 46 73 38 66 14 59)(3 28 47 74 39 67 15 60)(4 29 48 75 40 68 16 51)(5 30 49 76 31 69 17 52)(6 21 50 77 32 70 18 53)(7 22 41 78 33 61 19 54)(8 23 42 79 34 62 20 55)(9 24 43 80 35 63 11 56)(10 25 44 71 36 64 12 57)
G:=sub<Sym(80)| (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)(71,72,73,74,75,76,77,78,79,80), (1,3,5,7,9)(2,4,6,8,10)(11,19,17,15,13)(12,20,18,16,14)(21,27,23,29,25)(22,28,24,30,26)(31,33,35,37,39)(32,34,36,38,40)(41,49,47,45,43)(42,50,48,46,44)(51,55,59,53,57)(52,56,60,54,58)(61,67,63,69,65)(62,68,64,70,66)(71,75,79,73,77)(72,76,80,74,78), (1,26,45,72,37,65,13,58)(2,27,46,73,38,66,14,59)(3,28,47,74,39,67,15,60)(4,29,48,75,40,68,16,51)(5,30,49,76,31,69,17,52)(6,21,50,77,32,70,18,53)(7,22,41,78,33,61,19,54)(8,23,42,79,34,62,20,55)(9,24,43,80,35,63,11,56)(10,25,44,71,36,64,12,57)>;
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)(71,72,73,74,75,76,77,78,79,80), (1,3,5,7,9)(2,4,6,8,10)(11,19,17,15,13)(12,20,18,16,14)(21,27,23,29,25)(22,28,24,30,26)(31,33,35,37,39)(32,34,36,38,40)(41,49,47,45,43)(42,50,48,46,44)(51,55,59,53,57)(52,56,60,54,58)(61,67,63,69,65)(62,68,64,70,66)(71,75,79,73,77)(72,76,80,74,78), (1,26,45,72,37,65,13,58)(2,27,46,73,38,66,14,59)(3,28,47,74,39,67,15,60)(4,29,48,75,40,68,16,51)(5,30,49,76,31,69,17,52)(6,21,50,77,32,70,18,53)(7,22,41,78,33,61,19,54)(8,23,42,79,34,62,20,55)(9,24,43,80,35,63,11,56)(10,25,44,71,36,64,12,57) );
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),(71,72,73,74,75,76,77,78,79,80)], [(1,3,5,7,9),(2,4,6,8,10),(11,19,17,15,13),(12,20,18,16,14),(21,27,23,29,25),(22,28,24,30,26),(31,33,35,37,39),(32,34,36,38,40),(41,49,47,45,43),(42,50,48,46,44),(51,55,59,53,57),(52,56,60,54,58),(61,67,63,69,65),(62,68,64,70,66),(71,75,79,73,77),(72,76,80,74,78)], [(1,26,45,72,37,65,13,58),(2,27,46,73,38,66,14,59),(3,28,47,74,39,67,15,60),(4,29,48,75,40,68,16,51),(5,30,49,76,31,69,17,52),(6,21,50,77,32,70,18,53),(7,22,41,78,33,61,19,54),(8,23,42,79,34,62,20,55),(9,24,43,80,35,63,11,56),(10,25,44,71,36,64,12,57)]])
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 5E | ··· | 5I | 8A | ··· | 8H | 10A | ··· | 10L | 10M | ··· | 10AA | 20A | ··· | 20P | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 5 | ··· | 5 | 1 | ··· | 1 | 4 | ··· | 4 | 5 | ··· | 5 | 5 | ··· | 5 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | ||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C5 | C8 | C10 | C10 | C20 | C20 | C40 | F5 | C5⋊C8 | C2×F5 | C5×F5 | C5×C5⋊C8 | C10×F5 |
| kernel | C10×C5⋊C8 | C5×C5⋊C8 | C10×Dic5 | C5×Dic5 | C102 | C2×C5⋊C8 | C5×C10 | C5⋊C8 | C2×Dic5 | Dic5 | C2×C10 | C10 | C2×C10 | C10 | C10 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 8 | 8 | 4 | 8 | 8 | 32 | 1 | 2 | 1 | 4 | 8 | 4 |
Matrix representation of C10×C5⋊C8 ►in GL5(𝔽41)
| 37 | 0 | 0 | 0 | 0 |
| 0 | 31 | 0 | 0 | 0 |
| 0 | 0 | 31 | 0 | 0 |
| 0 | 0 | 0 | 31 | 0 |
| 0 | 0 | 0 | 0 | 31 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 | 0 |
| 0 | 19 | 16 | 0 | 0 |
| 0 | 18 | 0 | 37 | 0 |
| 0 | 4 | 0 | 0 | 10 |
| 3 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 26 | 0 |
| 0 | 9 | 0 | 1 | 40 |
| 0 | 17 | 40 | 25 | 0 |
| 0 | 38 | 0 | 13 | 0 |
G:=sub<GL(5,GF(41))| [37,0,0,0,0,0,31,0,0,0,0,0,31,0,0,0,0,0,31,0,0,0,0,0,31],[1,0,0,0,0,0,18,19,18,4,0,0,16,0,0,0,0,0,37,0,0,0,0,0,10],[3,0,0,0,0,0,16,9,17,38,0,0,0,40,0,0,26,1,25,13,0,0,40,0,0] >;
C10×C5⋊C8 in GAP, Magma, Sage, TeX
C_{10}\times C_5\rtimes C_8 % in TeX
G:=Group("C10xC5:C8"); // GroupNames label
G:=SmallGroup(400,139);
// by ID
G=gap.SmallGroup(400,139);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-2,-5,120,69,5765,599]);
// Polycyclic
G:=Group<a,b,c|a^10=b^5=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations
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