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## G = C5×C8⋊D5order 400 = 24·52

### Direct product of C5 and C8⋊D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C5×C8⋊D5
 Chief series C1 — C5 — C10 — C20 — C5×C20 — D5×C20 — C5×C8⋊D5
 Lower central C5 — C10 — C5×C8⋊D5
 Upper central C1 — C20 — C40

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 >

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 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 C4×D5 C5×D5 C8⋊D5 C5×M4(2) D5×C10 D5×C20 C5×C8⋊D5 kernel C5×C8⋊D5 C5×C5⋊2C8 C5×C40 D5×C20 C5×Dic5 D5×C10 C8⋊D5 C5⋊2C8 C40 C4×D5 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 C5×C8⋊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] >;

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

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