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## G = D5×C5×C10order 500 = 22·53

### Direct product of C5×C10 and D5

Aliases: D5×C5×C10, C5⋊C102, C534C22, C10⋊(C5×C10), (C5×C10)⋊3C10, C524(C2×C10), (C52×C10)⋊1C2, SmallGroup(500,53)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — D5×C5×C10
 Chief series C1 — C5 — C52 — C53 — D5×C52 — D5×C5×C10
 Lower central C5 — D5×C5×C10
 Upper central C1 — C5×C10

Generators and relations for D5×C5×C10
G = < a,b,c,d | a5=b10=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 272 in 128 conjugacy classes, 56 normal (10 characteristic)
C1, C2, C2, C22, C5, C5, C5, D5, C10, C10, C10, D10, C2×C10, C52, C52, C52, C5×D5, C5×C10, C5×C10, C5×C10, D5×C10, C102, C53, D5×C52, C52×C10, D5×C5×C10
Quotients: C1, C2, C22, C5, D5, C10, D10, C2×C10, C52, C5×D5, C5×C10, D5×C10, C102, D5×C52, D5×C5×C10

Smallest permutation representation of D5×C5×C10
On 100 points
Generators in S100
(1 54 35 47 20)(2 55 36 48 11)(3 56 37 49 12)(4 57 38 50 13)(5 58 39 41 14)(6 59 40 42 15)(7 60 31 43 16)(8 51 32 44 17)(9 52 33 45 18)(10 53 34 46 19)(21 81 98 66 73)(22 82 99 67 74)(23 83 100 68 75)(24 84 91 69 76)(25 85 92 70 77)(26 86 93 61 78)(27 87 94 62 79)(28 88 95 63 80)(29 89 96 64 71)(30 90 97 65 72)
(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)(81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100)
(1 5 9 3 7)(2 6 10 4 8)(11 15 19 13 17)(12 16 20 14 18)(21 27 23 29 25)(22 28 24 30 26)(31 35 39 33 37)(32 36 40 34 38)(41 45 49 43 47)(42 46 50 44 48)(51 55 59 53 57)(52 56 60 54 58)(61 67 63 69 65)(62 68 64 70 66)(71 77 73 79 75)(72 78 74 80 76)(81 87 83 89 85)(82 88 84 90 86)(91 97 93 99 95)(92 98 94 100 96)
(1 71)(2 72)(3 73)(4 74)(5 75)(6 76)(7 77)(8 78)(9 79)(10 80)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 61)(18 62)(19 63)(20 64)(21 56)(22 57)(23 58)(24 59)(25 60)(26 51)(27 52)(28 53)(29 54)(30 55)(31 85)(32 86)(33 87)(34 88)(35 89)(36 90)(37 81)(38 82)(39 83)(40 84)(41 100)(42 91)(43 92)(44 93)(45 94)(46 95)(47 96)(48 97)(49 98)(50 99)

G:=sub<Sym(100)| (1,54,35,47,20)(2,55,36,48,11)(3,56,37,49,12)(4,57,38,50,13)(5,58,39,41,14)(6,59,40,42,15)(7,60,31,43,16)(8,51,32,44,17)(9,52,33,45,18)(10,53,34,46,19)(21,81,98,66,73)(22,82,99,67,74)(23,83,100,68,75)(24,84,91,69,76)(25,85,92,70,77)(26,86,93,61,78)(27,87,94,62,79)(28,88,95,63,80)(29,89,96,64,71)(30,90,97,65,72), (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)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100), (1,5,9,3,7)(2,6,10,4,8)(11,15,19,13,17)(12,16,20,14,18)(21,27,23,29,25)(22,28,24,30,26)(31,35,39,33,37)(32,36,40,34,38)(41,45,49,43,47)(42,46,50,44,48)(51,55,59,53,57)(52,56,60,54,58)(61,67,63,69,65)(62,68,64,70,66)(71,77,73,79,75)(72,78,74,80,76)(81,87,83,89,85)(82,88,84,90,86)(91,97,93,99,95)(92,98,94,100,96), (1,71)(2,72)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,79)(10,80)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,61)(18,62)(19,63)(20,64)(21,56)(22,57)(23,58)(24,59)(25,60)(26,51)(27,52)(28,53)(29,54)(30,55)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(37,81)(38,82)(39,83)(40,84)(41,100)(42,91)(43,92)(44,93)(45,94)(46,95)(47,96)(48,97)(49,98)(50,99)>;

G:=Group( (1,54,35,47,20)(2,55,36,48,11)(3,56,37,49,12)(4,57,38,50,13)(5,58,39,41,14)(6,59,40,42,15)(7,60,31,43,16)(8,51,32,44,17)(9,52,33,45,18)(10,53,34,46,19)(21,81,98,66,73)(22,82,99,67,74)(23,83,100,68,75)(24,84,91,69,76)(25,85,92,70,77)(26,86,93,61,78)(27,87,94,62,79)(28,88,95,63,80)(29,89,96,64,71)(30,90,97,65,72), (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)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100), (1,5,9,3,7)(2,6,10,4,8)(11,15,19,13,17)(12,16,20,14,18)(21,27,23,29,25)(22,28,24,30,26)(31,35,39,33,37)(32,36,40,34,38)(41,45,49,43,47)(42,46,50,44,48)(51,55,59,53,57)(52,56,60,54,58)(61,67,63,69,65)(62,68,64,70,66)(71,77,73,79,75)(72,78,74,80,76)(81,87,83,89,85)(82,88,84,90,86)(91,97,93,99,95)(92,98,94,100,96), (1,71)(2,72)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,79)(10,80)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,61)(18,62)(19,63)(20,64)(21,56)(22,57)(23,58)(24,59)(25,60)(26,51)(27,52)(28,53)(29,54)(30,55)(31,85)(32,86)(33,87)(34,88)(35,89)(36,90)(37,81)(38,82)(39,83)(40,84)(41,100)(42,91)(43,92)(44,93)(45,94)(46,95)(47,96)(48,97)(49,98)(50,99) );

G=PermutationGroup([[(1,54,35,47,20),(2,55,36,48,11),(3,56,37,49,12),(4,57,38,50,13),(5,58,39,41,14),(6,59,40,42,15),(7,60,31,43,16),(8,51,32,44,17),(9,52,33,45,18),(10,53,34,46,19),(21,81,98,66,73),(22,82,99,67,74),(23,83,100,68,75),(24,84,91,69,76),(25,85,92,70,77),(26,86,93,61,78),(27,87,94,62,79),(28,88,95,63,80),(29,89,96,64,71),(30,90,97,65,72)], [(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),(81,82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99,100)], [(1,5,9,3,7),(2,6,10,4,8),(11,15,19,13,17),(12,16,20,14,18),(21,27,23,29,25),(22,28,24,30,26),(31,35,39,33,37),(32,36,40,34,38),(41,45,49,43,47),(42,46,50,44,48),(51,55,59,53,57),(52,56,60,54,58),(61,67,63,69,65),(62,68,64,70,66),(71,77,73,79,75),(72,78,74,80,76),(81,87,83,89,85),(82,88,84,90,86),(91,97,93,99,95),(92,98,94,100,96)], [(1,71),(2,72),(3,73),(4,74),(5,75),(6,76),(7,77),(8,78),(9,79),(10,80),(11,65),(12,66),(13,67),(14,68),(15,69),(16,70),(17,61),(18,62),(19,63),(20,64),(21,56),(22,57),(23,58),(24,59),(25,60),(26,51),(27,52),(28,53),(29,54),(30,55),(31,85),(32,86),(33,87),(34,88),(35,89),(36,90),(37,81),(38,82),(39,83),(40,84),(41,100),(42,91),(43,92),(44,93),(45,94),(46,95),(47,96),(48,97),(49,98),(50,99)]])

200 conjugacy classes

 class 1 2A 2B 2C 5A ··· 5X 5Y ··· 5BV 10A ··· 10X 10Y ··· 10BV 10BW ··· 10DR order 1 2 2 2 5 ··· 5 5 ··· 5 10 ··· 10 10 ··· 10 10 ··· 10 size 1 1 5 5 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 5 ··· 5

200 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C5 C10 C10 D5 D10 C5×D5 D5×C10 kernel D5×C5×C10 D5×C52 C52×C10 D5×C10 C5×D5 C5×C10 C5×C10 C52 C10 C5 # reps 1 2 1 24 48 24 2 2 48 48

Matrix representation of D5×C5×C10 in GL3(𝔽11) generated by

 9 0 0 0 9 0 0 0 9
,
 2 0 0 0 4 0 0 0 4
,
 1 0 0 0 3 0 0 0 4
,
 10 0 0 0 0 4 0 3 0
G:=sub<GL(3,GF(11))| [9,0,0,0,9,0,0,0,9],[2,0,0,0,4,0,0,0,4],[1,0,0,0,3,0,0,0,4],[10,0,0,0,0,3,0,4,0] >;

D5×C5×C10 in GAP, Magma, Sage, TeX

D_5\times C_5\times C_{10}
% in TeX

G:=Group("D5xC5xC10");
// GroupNames label

G:=SmallGroup(500,53);
// by ID

G=gap.SmallGroup(500,53);
# by ID

G:=PCGroup([5,-2,-2,-5,-5,-5,10004]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^10=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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