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## G = D5×C5⋊2C8order 400 = 24·52

### Direct product of D5 and C5⋊2C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — D5×C5⋊2C8
 Chief series C1 — C5 — C52 — C5×C10 — C5×C20 — D5×C20 — D5×C5⋊2C8
 Lower central C52 — D5×C5⋊2C8
 Upper central C1 — C4

Generators and relations for D5×C52C8
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×C52C8
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×C52C8 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×C52C8 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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