direct product, metabelian, supersoluble, monomial, A-group
Aliases: D5×C5⋊2C8, C20.28D10, D10.4Dic5, Dic5.4Dic5, C5⋊5(C8×D5), C4.13D52, (C5×D5)⋊4C8, C52⋊9(C2×C8), (C4×D5).7D5, C52⋊7C8⋊6C2, (D5×C20).6C2, (D5×C10).5C4, C10.23(C4×D5), C2.1(D5×Dic5), (C5×Dic5).8C4, C10.8(C2×Dic5), (C5×C20).27C22, C5⋊3(C2×C5⋊2C8), (C5×C5⋊2C8)⋊4C2, (C5×C10).41(C2×C4), SmallGroup(400,60)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — D5×C5⋊2C8 |
Generators and relations for D5×C5⋊2C8
G = < a,b,c,d | a5=b2=c5=d8=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Smallest permutation representation of D5×C5⋊2C8
►On 80 points
Generators in S80
(1 59 71 44 53)(2 60 72 45 54)(3 61 65 46 55)(4 62 66 47 56)(5 63 67 48 49)(6 64 68 41 50)(7 57 69 42 51)(8 58 70 43 52)(9 35 19 29 76)(10 36 20 30 77)(11 37 21 31 78)(12 38 22 32 79)(13 39 23 25 80)(14 40 24 26 73)(15 33 17 27 74)(16 34 18 28 75)
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 33)(8 34)(9 59)(10 60)(11 61)(12 62)(13 63)(14 64)(15 57)(16 58)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 49)(24 50)(25 48)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(65 78)(66 79)(67 80)(68 73)(69 74)(70 75)(71 76)(72 77)
(1 71 53 59 44)(2 45 60 54 72)(3 65 55 61 46)(4 47 62 56 66)(5 67 49 63 48)(6 41 64 50 68)(7 69 51 57 42)(8 43 58 52 70)(9 29 35 76 19)(10 20 77 36 30)(11 31 37 78 21)(12 22 79 38 32)(13 25 39 80 23)(14 24 73 40 26)(15 27 33 74 17)(16 18 75 34 28)
(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)
G:=sub<Sym(80)| (1,59,71,44,53)(2,60,72,45,54)(3,61,65,46,55)(4,62,66,47,56)(5,63,67,48,49)(6,64,68,41,50)(7,57,69,42,51)(8,58,70,43,52)(9,35,19,29,76)(10,36,20,30,77)(11,37,21,31,78)(12,38,22,32,79)(13,39,23,25,80)(14,40,24,26,73)(15,33,17,27,74)(16,34,18,28,75), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,33)(8,34)(9,59)(10,60)(11,61)(12,62)(13,63)(14,64)(15,57)(16,58)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,49)(24,50)(25,48)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(65,78)(66,79)(67,80)(68,73)(69,74)(70,75)(71,76)(72,77), (1,71,53,59,44)(2,45,60,54,72)(3,65,55,61,46)(4,47,62,56,66)(5,67,49,63,48)(6,41,64,50,68)(7,69,51,57,42)(8,43,58,52,70)(9,29,35,76,19)(10,20,77,36,30)(11,31,37,78,21)(12,22,79,38,32)(13,25,39,80,23)(14,24,73,40,26)(15,27,33,74,17)(16,18,75,34,28), (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)>;
G:=Group( (1,59,71,44,53)(2,60,72,45,54)(3,61,65,46,55)(4,62,66,47,56)(5,63,67,48,49)(6,64,68,41,50)(7,57,69,42,51)(8,58,70,43,52)(9,35,19,29,76)(10,36,20,30,77)(11,37,21,31,78)(12,38,22,32,79)(13,39,23,25,80)(14,40,24,26,73)(15,33,17,27,74)(16,34,18,28,75), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,33)(8,34)(9,59)(10,60)(11,61)(12,62)(13,63)(14,64)(15,57)(16,58)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,49)(24,50)(25,48)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(65,78)(66,79)(67,80)(68,73)(69,74)(70,75)(71,76)(72,77), (1,71,53,59,44)(2,45,60,54,72)(3,65,55,61,46)(4,47,62,56,66)(5,67,49,63,48)(6,41,64,50,68)(7,69,51,57,42)(8,43,58,52,70)(9,29,35,76,19)(10,20,77,36,30)(11,31,37,78,21)(12,22,79,38,32)(13,25,39,80,23)(14,24,73,40,26)(15,27,33,74,17)(16,18,75,34,28), (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) );
G=PermutationGroup([[(1,59,71,44,53),(2,60,72,45,54),(3,61,65,46,55),(4,62,66,47,56),(5,63,67,48,49),(6,64,68,41,50),(7,57,69,42,51),(8,58,70,43,52),(9,35,19,29,76),(10,36,20,30,77),(11,37,21,31,78),(12,38,22,32,79),(13,39,23,25,80),(14,40,24,26,73),(15,33,17,27,74),(16,34,18,28,75)], [(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,33),(8,34),(9,59),(10,60),(11,61),(12,62),(13,63),(14,64),(15,57),(16,58),(17,51),(18,52),(19,53),(20,54),(21,55),(22,56),(23,49),(24,50),(25,48),(26,41),(27,42),(28,43),(29,44),(30,45),(31,46),(32,47),(65,78),(66,79),(67,80),(68,73),(69,74),(70,75),(71,76),(72,77)], [(1,71,53,59,44),(2,45,60,54,72),(3,65,55,61,46),(4,47,62,56,66),(5,67,49,63,48),(6,41,64,50,68),(7,69,51,57,42),(8,43,58,52,70),(9,29,35,76,19),(10,20,77,36,30),(11,31,37,78,21),(12,22,79,38,32),(13,25,39,80,23),(14,24,73,40,26),(15,27,33,74,17),(16,18,75,34,28)], [(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)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 5E | 5F | 5G | 5H | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 20A | ··· | 20H | 20I | ··· | 20P | 20Q | 20R | 20S | 20T | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 5 | 5 | 1 | 1 | 5 | 5 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 25 | 25 | 25 | 25 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | ··· | 4 | 10 | 10 | 10 | 10 | 10 | ··· | 10 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D5 | D5 | Dic5 | D10 | Dic5 | C5⋊2C8 | C4×D5 | C8×D5 | D52 | D5×Dic5 | D5×C5⋊2C8 |
| kernel | D5×C5⋊2C8 | C5×C5⋊2C8 | C52⋊7C8 | D5×C20 | C5×Dic5 | D5×C10 | C5×D5 | C5⋊2C8 | C4×D5 | Dic5 | C20 | D10 | D5 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 4 | 2 | 8 | 4 | 8 | 4 | 4 | 8 |
Matrix representation of D5×C5⋊2C8 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 34 | 40 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 7 | 1 |
| 0 | 0 | 34 | 34 |
| 34 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 27 | 0 | 0 |
| 27 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,34,1,0,0,40,0],[1,0,0,0,0,1,0,0,0,0,7,34,0,0,1,34],[34,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,27,0,0,27,0,0,0,0,0,9,0,0,0,0,9] >;
D5×C5⋊2C8 in GAP, Magma, Sage, TeX
D_5\times C_5\rtimes_2C_8
% in TeX
G:=Group("D5xC5:2C8"); // GroupNames label
G:=SmallGroup(400,60);
// by ID
G=gap.SmallGroup(400,60);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,31,50,970,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^5=d^8=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
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