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G = D5xC5xC10order 500 = 22·53

Direct product of C5xC10 and D5

direct product, metabelian, supersoluble, monomial, A-group

Aliases: D5xC5xC10, C5:C102, C53:4C22, C10:(C5xC10), (C5xC10):3C10, C52:4(C2xC10), (C52xC10):1C2, SmallGroup(500,53)

Series: Derived Chief Lower central Upper central

C1C5 — D5xC5xC10
C1C5C52C53D5xC52 — D5xC5xC10
C5 — D5xC5xC10
C1C5xC10

Generators and relations for D5xC5xC10
 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, C2xC10, C52, C52, C52, C5xD5, C5xC10, C5xC10, C5xC10, D5xC10, C102, C53, D5xC52, C52xC10, D5xC5xC10
Quotients: C1, C2, C22, C5, D5, C10, D10, C2xC10, C52, C5xD5, C5xC10, D5xC10, C102, D5xC52, D5xC5xC10

Smallest permutation representation of D5xC5xC10
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 2A2B2C5A···5X5Y···5BV10A···10X10Y···10BV10BW···10DR
order12225···55···510···1010···1010···10
size11551···12···21···12···25···5

200 irreducible representations

dim1111112222
type+++++
imageC1C2C2C5C10C10D5D10C5xD5D5xC10
kernelD5xC5xC10D5xC52C52xC10D5xC10C5xD5C5xC10C5xC10C52C10C5
# reps121244824224848

Matrix representation of D5xC5xC10 in GL3(F11) generated by

900
090
009
,
200
040
004
,
100
030
004
,
1000
004
030
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] >;

D5xC5xC10 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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