direct product, metacyclic, supersoluble, monomial
Aliases: C5xC8:D5, C40:7D5, C40:4C10, D10.1C20, C20.65D10, Dic5.1C20, C52:15M4(2), C8:3(C5xD5), (C5xC40):8C2, C5:2C8:4C10, C2.3(D5xC20), C10.9(C2xC20), (C4xD5).2C10, (D5xC10).7C4, (D5xC20).7C2, C10.29(C4xD5), C4.13(D5xC10), C5:3(C5xM4(2)), C20.14(C2xC10), (C5xC20).43C22, (C5xDic5).10C4, (C5xC5:2C8):11C2, (C5xC10).51(C2xC4), SmallGroup(400,77)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5xC8:D5
G = < a,b,c,d | a5=b8=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >
Quotients: C1, C2, C4, C22, C5, C2xC4, D5, C10, M4(2), C20, D10, C2xC10, C4xD5, C2xC20, C5xD5, C8:D5, C5xM4(2), D5xC10, D5xC20, C5xC8:D5
Smallest permutation representation of C5xC8:D5
►On 80 points
Generators in S80
(1 65 59 46 54)(2 66 60 47 55)(3 67 61 48 56)(4 68 62 41 49)(5 69 63 42 50)(6 70 64 43 51)(7 71 57 44 52)(8 72 58 45 53)(9 35 28 24 80)(10 36 29 17 73)(11 37 30 18 74)(12 38 31 19 75)(13 39 32 20 76)(14 40 25 21 77)(15 33 26 22 78)(16 34 27 23 79)
(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 65 59 46 54)(2 66 60 47 55)(3 67 61 48 56)(4 68 62 41 49)(5 69 63 42 50)(6 70 64 43 51)(7 71 57 44 52)(8 72 58 45 53)(9 80 24 28 35)(10 73 17 29 36)(11 74 18 30 37)(12 75 19 31 38)(13 76 20 32 39)(14 77 21 25 40)(15 78 22 26 33)(16 79 23 27 34)
(1 20)(2 17)(3 22)(4 19)(5 24)(6 21)(7 18)(8 23)(9 63)(10 60)(11 57)(12 62)(13 59)(14 64)(15 61)(16 58)(25 51)(26 56)(27 53)(28 50)(29 55)(30 52)(31 49)(32 54)(33 48)(34 45)(35 42)(36 47)(37 44)(38 41)(39 46)(40 43)(65 76)(66 73)(67 78)(68 75)(69 80)(70 77)(71 74)(72 79)
G:=sub<Sym(80)| (1,65,59,46,54)(2,66,60,47,55)(3,67,61,48,56)(4,68,62,41,49)(5,69,63,42,50)(6,70,64,43,51)(7,71,57,44,52)(8,72,58,45,53)(9,35,28,24,80)(10,36,29,17,73)(11,37,30,18,74)(12,38,31,19,75)(13,39,32,20,76)(14,40,25,21,77)(15,33,26,22,78)(16,34,27,23,79), (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), (1,65,59,46,54)(2,66,60,47,55)(3,67,61,48,56)(4,68,62,41,49)(5,69,63,42,50)(6,70,64,43,51)(7,71,57,44,52)(8,72,58,45,53)(9,80,24,28,35)(10,73,17,29,36)(11,74,18,30,37)(12,75,19,31,38)(13,76,20,32,39)(14,77,21,25,40)(15,78,22,26,33)(16,79,23,27,34), (1,20)(2,17)(3,22)(4,19)(5,24)(6,21)(7,18)(8,23)(9,63)(10,60)(11,57)(12,62)(13,59)(14,64)(15,61)(16,58)(25,51)(26,56)(27,53)(28,50)(29,55)(30,52)(31,49)(32,54)(33,48)(34,45)(35,42)(36,47)(37,44)(38,41)(39,46)(40,43)(65,76)(66,73)(67,78)(68,75)(69,80)(70,77)(71,74)(72,79)>;
G:=Group( (1,65,59,46,54)(2,66,60,47,55)(3,67,61,48,56)(4,68,62,41,49)(5,69,63,42,50)(6,70,64,43,51)(7,71,57,44,52)(8,72,58,45,53)(9,35,28,24,80)(10,36,29,17,73)(11,37,30,18,74)(12,38,31,19,75)(13,39,32,20,76)(14,40,25,21,77)(15,33,26,22,78)(16,34,27,23,79), (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), (1,65,59,46,54)(2,66,60,47,55)(3,67,61,48,56)(4,68,62,41,49)(5,69,63,42,50)(6,70,64,43,51)(7,71,57,44,52)(8,72,58,45,53)(9,80,24,28,35)(10,73,17,29,36)(11,74,18,30,37)(12,75,19,31,38)(13,76,20,32,39)(14,77,21,25,40)(15,78,22,26,33)(16,79,23,27,34), (1,20)(2,17)(3,22)(4,19)(5,24)(6,21)(7,18)(8,23)(9,63)(10,60)(11,57)(12,62)(13,59)(14,64)(15,61)(16,58)(25,51)(26,56)(27,53)(28,50)(29,55)(30,52)(31,49)(32,54)(33,48)(34,45)(35,42)(36,47)(37,44)(38,41)(39,46)(40,43)(65,76)(66,73)(67,78)(68,75)(69,80)(70,77)(71,74)(72,79) );
G=PermutationGroup([[(1,65,59,46,54),(2,66,60,47,55),(3,67,61,48,56),(4,68,62,41,49),(5,69,63,42,50),(6,70,64,43,51),(7,71,57,44,52),(8,72,58,45,53),(9,35,28,24,80),(10,36,29,17,73),(11,37,30,18,74),(12,38,31,19,75),(13,39,32,20,76),(14,40,25,21,77),(15,33,26,22,78),(16,34,27,23,79)], [(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)], [(1,65,59,46,54),(2,66,60,47,55),(3,67,61,48,56),(4,68,62,41,49),(5,69,63,42,50),(6,70,64,43,51),(7,71,57,44,52),(8,72,58,45,53),(9,80,24,28,35),(10,73,17,29,36),(11,74,18,30,37),(12,75,19,31,38),(13,76,20,32,39),(14,77,21,25,40),(15,78,22,26,33),(16,79,23,27,34)], [(1,20),(2,17),(3,22),(4,19),(5,24),(6,21),(7,18),(8,23),(9,63),(10,60),(11,57),(12,62),(13,59),(14,64),(15,61),(16,58),(25,51),(26,56),(27,53),(28,50),(29,55),(30,52),(31,49),(32,54),(33,48),(34,45),(35,42),(36,47),(37,44),(38,41),(39,46),(40,43),(65,76),(66,73),(67,78),(68,75),(69,80),(70,77),(71,74),(72,79)]])
130 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 5E | ··· | 5N | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | ··· | 10N | 10O | 10P | 10Q | 10R | 20A | ··· | 20H | 20I | ··· | 20AB | 20AC | 20AD | 20AE | 20AF | 40A | ··· | 40AV | 40AW | ··· | 40BD |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 5 | ··· | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 10 | 1 | 1 | 10 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 10 | 10 | 10 | 10 | 1 | ··· | 1 | 2 | ··· | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 10 | ··· | 10 |
130 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C5 | C10 | C10 | C10 | C20 | C20 | D5 | M4(2) | D10 | C4xD5 | C5xD5 | C8:D5 | C5xM4(2) | D5xC10 | D5xC20 | C5xC8:D5 |
| kernel | C5xC8:D5 | C5xC5:2C8 | C5xC40 | D5xC20 | C5xDic5 | D5xC10 | C8:D5 | C5:2C8 | C40 | C4xD5 | Dic5 | D10 | C40 | C52 | C20 | C10 | C8 | C5 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 4 | 8 | 8 | 8 | 8 | 16 | 32 |
Matrix representation of C5xC8:D5 ►in GL2(F41) generated by
| 10 | 0 |
| 0 | 10 |
| 27 | 0 |
| 0 | 14 |
| 10 | 0 |
| 0 | 37 |
| 0 | 37 |
| 10 | 0 |
G:=sub<GL(2,GF(41))| [10,0,0,10],[27,0,0,14],[10,0,0,37],[0,10,37,0] >;
C5xC8:D5 in GAP, Magma, Sage, TeX
C_5\times C_8\rtimes D_5
% in TeX
G:=Group("C5xC8:D5"); // GroupNames label
G:=SmallGroup(400,77);
// by ID
G=gap.SmallGroup(400,77);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-2,-5,505,127,69,11525]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations
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