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## G = D5×C5⋊C8order 400 = 24·52

### Direct product of D5 and C5⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — D5×C5⋊C8
 Chief series C1 — C5 — C52 — C5×C10 — C5×Dic5 — C5×C5⋊C8 — D5×C5⋊C8
 Lower central C52 — D5×C5⋊C8
 Upper central C1 — C2

Generators and relations for D5×C5⋊C8
G = < a,b,c,d | a5=b2=c5=d8=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Smallest permutation representation of D5×C5⋊C8
On 80 points
Generators in S80
(1 33 75 19 57)(2 34 76 20 58)(3 35 77 21 59)(4 36 78 22 60)(5 37 79 23 61)(6 38 80 24 62)(7 39 73 17 63)(8 40 74 18 64)(9 71 53 45 29)(10 72 54 46 30)(11 65 55 47 31)(12 66 56 48 32)(13 67 49 41 25)(14 68 50 42 26)(15 69 51 43 27)(16 70 52 44 28)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)(17 53)(18 54)(19 55)(20 56)(21 49)(22 50)(23 51)(24 52)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(31 33)(32 34)(41 77)(42 78)(43 79)(44 80)(45 73)(46 74)(47 75)(48 76)(57 65)(58 66)(59 67)(60 68)(61 69)(62 70)(63 71)(64 72)
(1 33 75 19 57)(2 20 34 58 76)(3 59 21 77 35)(4 78 60 36 22)(5 37 79 23 61)(6 24 38 62 80)(7 63 17 73 39)(8 74 64 40 18)(9 71 53 45 29)(10 46 72 30 54)(11 31 47 55 65)(12 56 32 66 48)(13 67 49 41 25)(14 42 68 26 50)(15 27 43 51 69)(16 52 28 70 44)
(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)| (1,33,75,19,57)(2,34,76,20,58)(3,35,77,21,59)(4,36,78,22,60)(5,37,79,23,61)(6,38,80,24,62)(7,39,73,17,63)(8,40,74,18,64)(9,71,53,45,29)(10,72,54,46,30)(11,65,55,47,31)(12,66,56,48,32)(13,67,49,41,25)(14,68,50,42,26)(15,69,51,43,27)(16,70,52,44,28), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,53)(18,54)(19,55)(20,56)(21,49)(22,50)(23,51)(24,52)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(31,33)(32,34)(41,77)(42,78)(43,79)(44,80)(45,73)(46,74)(47,75)(48,76)(57,65)(58,66)(59,67)(60,68)(61,69)(62,70)(63,71)(64,72), (1,33,75,19,57)(2,20,34,58,76)(3,59,21,77,35)(4,78,60,36,22)(5,37,79,23,61)(6,24,38,62,80)(7,63,17,73,39)(8,74,64,40,18)(9,71,53,45,29)(10,46,72,30,54)(11,31,47,55,65)(12,56,32,66,48)(13,67,49,41,25)(14,42,68,26,50)(15,27,43,51,69)(16,52,28,70,44), (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( (1,33,75,19,57)(2,34,76,20,58)(3,35,77,21,59)(4,36,78,22,60)(5,37,79,23,61)(6,38,80,24,62)(7,39,73,17,63)(8,40,74,18,64)(9,71,53,45,29)(10,72,54,46,30)(11,65,55,47,31)(12,66,56,48,32)(13,67,49,41,25)(14,68,50,42,26)(15,69,51,43,27)(16,70,52,44,28), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)(17,53)(18,54)(19,55)(20,56)(21,49)(22,50)(23,51)(24,52)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(31,33)(32,34)(41,77)(42,78)(43,79)(44,80)(45,73)(46,74)(47,75)(48,76)(57,65)(58,66)(59,67)(60,68)(61,69)(62,70)(63,71)(64,72), (1,33,75,19,57)(2,20,34,58,76)(3,59,21,77,35)(4,78,60,36,22)(5,37,79,23,61)(6,24,38,62,80)(7,63,17,73,39)(8,74,64,40,18)(9,71,53,45,29)(10,46,72,30,54)(11,31,47,55,65)(12,56,32,66,48)(13,67,49,41,25)(14,42,68,26,50)(15,27,43,51,69)(16,52,28,70,44), (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([[(1,33,75,19,57),(2,34,76,20,58),(3,35,77,21,59),(4,36,78,22,60),(5,37,79,23,61),(6,38,80,24,62),(7,39,73,17,63),(8,40,74,18,64),(9,71,53,45,29),(10,72,54,46,30),(11,65,55,47,31),(12,66,56,48,32),(13,67,49,41,25),(14,68,50,42,26),(15,69,51,43,27),(16,70,52,44,28)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,9),(8,10),(17,53),(18,54),(19,55),(20,56),(21,49),(22,50),(23,51),(24,52),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(31,33),(32,34),(41,77),(42,78),(43,79),(44,80),(45,73),(46,74),(47,75),(48,76),(57,65),(58,66),(59,67),(60,68),(61,69),(62,70),(63,71),(64,72)], [(1,33,75,19,57),(2,20,34,58,76),(3,59,21,77,35),(4,78,60,36,22),(5,37,79,23,61),(6,24,38,62,80),(7,63,17,73,39),(8,74,64,40,18),(9,71,53,45,29),(10,46,72,30,54),(11,31,47,55,65),(12,56,32,66,48),(13,67,49,41,25),(14,42,68,26,50),(15,27,43,51,69),(16,52,28,70,44)], [(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)]])

40 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 5C 5D 5E 8A 8B 8C 8D 8E 8F 8G 8H 10A 10B 10C 10D 10E 10F 10G 20A 20B 20C 20D 40A ··· 40H order 1 2 2 2 4 4 4 4 5 5 5 5 5 8 8 8 8 8 8 8 8 10 10 10 10 10 10 10 20 20 20 20 40 ··· 40 size 1 1 5 5 5 5 25 25 2 2 4 8 8 5 5 5 5 25 25 25 25 2 2 4 8 8 20 20 10 10 10 10 10 ··· 10

40 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 4 4 4 8 8 type + + + + + + + - + + - image C1 C2 C2 C2 C4 C4 C8 D5 D10 C4×D5 C8×D5 F5 C5⋊C8 C2×F5 D5×F5 D5×C5⋊C8 kernel D5×C5⋊C8 C5×C5⋊C8 C52⋊3C8 D5×Dic5 C52⋊6C4 D5×C10 C5×D5 C5⋊C8 Dic5 C10 C5 D10 D5 C10 C2 C1 # reps 1 1 1 1 2 2 8 2 2 4 8 1 2 1 2 2

Matrix representation of D5×C5⋊C8 in GL6(𝔽41)

 0 40 0 0 0 0 1 34 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 40 7 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 40 40 40 40 0 0 1 0 0 0
,
 3 0 0 0 0 0 0 3 0 0 0 0 0 0 1 0 0 0 0 0 40 40 40 40 0 0 0 1 0 0 0 0 0 0 1 0

G:=sub<GL(6,GF(41))| [0,1,0,0,0,0,40,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,7,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,1,0,0,1,0,40,0,0,0,0,1,40,0,0,0,0,0,40,0],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,40,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0] >;

D5×C5⋊C8 in GAP, Magma, Sage, TeX

D_5\times C_5\rtimes C_8
% in TeX

G:=Group("D5xC5:C8");
// GroupNames label

G:=SmallGroup(400,120);
// by ID

G=gap.SmallGroup(400,120);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,50,970,5765,2897]);
// 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^3>;
// generators/relations

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