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G = C5×C8⋊D5order 400 = 24·52

Direct product of C5 and C8⋊D5

direct product, metacyclic, supersoluble, monomial

Aliases: C5×C8⋊D5, C407D5, C404C10, D10.1C20, C20.65D10, Dic5.1C20, C5215M4(2), C83(C5×D5), (C5×C40)⋊8C2, C52C84C10, C2.3(D5×C20), C10.9(C2×C20), (C4×D5).2C10, (D5×C10).7C4, (D5×C20).7C2, C10.29(C4×D5), C4.13(D5×C10), C53(C5×M4(2)), C20.14(C2×C10), (C5×C20).43C22, (C5×Dic5).10C4, (C5×C52C8)⋊11C2, (C5×C10).51(C2×C4), SmallGroup(400,77)

Series: Derived Chief Lower central Upper central

C1C10 — C5×C8⋊D5
C1C5C10C20C5×C20D5×C20 — C5×C8⋊D5
C5C10 — C5×C8⋊D5
C1C20C40

Generators and relations for C5×C8⋊D5
 G = < a,b,c,d | a5=b8=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >

10C2
2C5
2C5
5C4
5C22
2C10
2C10
2D5
10C10
5C2×C4
5C8
2C20
2C20
5C20
5C2×C10
2C5×D5
5M4(2)
2C40
2C40
5C2×C20
5C40
5C5×M4(2)

Smallest permutation representation of C5×C8⋊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 78 25 33 21)(10 79 26 34 22)(11 80 27 35 23)(12 73 28 36 24)(13 74 29 37 17)(14 75 30 38 18)(15 76 31 39 19)(16 77 32 40 20)
(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 21 33 25 78)(10 22 34 26 79)(11 23 35 27 80)(12 24 36 28 73)(13 17 37 29 74)(14 18 38 30 75)(15 19 39 31 76)(16 20 40 32 77)
(1 15)(2 12)(3 9)(4 14)(5 11)(6 16)(7 13)(8 10)(17 52)(18 49)(19 54)(20 51)(21 56)(22 53)(23 50)(24 55)(25 61)(26 58)(27 63)(28 60)(29 57)(30 62)(31 59)(32 64)(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)(9,78,25,33,21)(10,79,26,34,22)(11,80,27,35,23)(12,73,28,36,24)(13,74,29,37,17)(14,75,30,38,18)(15,76,31,39,19)(16,77,32,40,20), (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)(9,21,33,25,78)(10,22,34,26,79)(11,23,35,27,80)(12,24,36,28,73)(13,17,37,29,74)(14,18,38,30,75)(15,19,39,31,76)(16,20,40,32,77), (1,15)(2,12)(3,9)(4,14)(5,11)(6,16)(7,13)(8,10)(17,52)(18,49)(19,54)(20,51)(21,56)(22,53)(23,50)(24,55)(25,61)(26,58)(27,63)(28,60)(29,57)(30,62)(31,59)(32,64)(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)(9,78,25,33,21)(10,79,26,34,22)(11,80,27,35,23)(12,73,28,36,24)(13,74,29,37,17)(14,75,30,38,18)(15,76,31,39,19)(16,77,32,40,20), (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)(9,21,33,25,78)(10,22,34,26,79)(11,23,35,27,80)(12,24,36,28,73)(13,17,37,29,74)(14,18,38,30,75)(15,19,39,31,76)(16,20,40,32,77), (1,15)(2,12)(3,9)(4,14)(5,11)(6,16)(7,13)(8,10)(17,52)(18,49)(19,54)(20,51)(21,56)(22,53)(23,50)(24,55)(25,61)(26,58)(27,63)(28,60)(29,57)(30,62)(31,59)(32,64)(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),(9,78,25,33,21),(10,79,26,34,22),(11,80,27,35,23),(12,73,28,36,24),(13,74,29,37,17),(14,75,30,38,18),(15,76,31,39,19),(16,77,32,40,20)], [(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),(9,21,33,25,78),(10,22,34,26,79),(11,23,35,27,80),(12,24,36,28,73),(13,17,37,29,74),(14,18,38,30,75),(15,19,39,31,76),(16,20,40,32,77)], [(1,15),(2,12),(3,9),(4,14),(5,11),(6,16),(7,13),(8,10),(17,52),(18,49),(19,54),(20,51),(21,56),(22,53),(23,50),(24,55),(25,61),(26,58),(27,63),(28,60),(29,57),(30,62),(31,59),(32,64),(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 2A2B4A4B4C5A5B5C5D5E···5N8A8B8C8D10A10B10C10D10E···10N10O10P10Q10R20A···20H20I···20AB20AC20AD20AE20AF40A···40AV40AW···40BD
order12244455555···588881010101010···101010101020···2020···202020202040···4040···40
size1110111011112···222101011112···2101010101···12···2101010102···210···10

130 irreducible representations

dim1111111111112222222222
type++++++
imageC1C2C2C2C4C4C5C10C10C10C20C20D5M4(2)D10C4×D5C5×D5C8⋊D5C5×M4(2)D5×C10D5×C20C5×C8⋊D5
kernelC5×C8⋊D5C5×C52C8C5×C40D5×C20C5×Dic5D5×C10C8⋊D5C52C8C40C4×D5Dic5D10C40C52C20C10C8C5C5C4C2C1
# reps111122444488222488881632

Matrix representation of C5×C8⋊D5 in GL2(𝔽41) generated by

100
010
,
270
014
,
100
037
,
037
100
G:=sub<GL(2,GF(41))| [10,0,0,10],[27,0,0,14],[10,0,0,37],[0,10,37,0] >;

C5×C8⋊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

Export

Subgroup lattice of C5×C8⋊D5 in TeX

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