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## G = Dic5×C52order 500 = 22·53

### Direct product of C52 and Dic5

Aliases: Dic5×C52, C5310C4, C528C20, C52(C5×C20), C10.(C5×C10), C2.(D5×C52), (C5×C10).7C10, C10.10(C5×D5), (C5×C10).11D5, (C52×C10).1C2, SmallGroup(500,37)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — Dic5×C52
 Chief series C1 — C5 — C10 — C5×C10 — C52×C10 — Dic5×C52
 Lower central C5 — Dic5×C52
 Upper central C1 — C5×C10

Generators and relations for Dic5×C52
G = < a,b,c,d | a5=b5=c10=1, d2=c5, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 176 in 96 conjugacy classes, 40 normal (10 characteristic)
C1, C2, C4, C5, C5, C5, C10, C10, C10, Dic5, C20, C52, C52, C52, C5×C10, C5×C10, C5×C10, C5×Dic5, C5×C20, C53, C52×C10, Dic5×C52
Quotients: C1, C2, C4, C5, D5, C10, Dic5, C20, C52, C5×D5, C5×C10, C5×Dic5, C5×C20, D5×C52, Dic5×C52

Smallest permutation representation of Dic5×C52
On 100 points
Generators in S100
(1 37 15 43 21)(2 38 16 44 22)(3 39 17 45 23)(4 40 18 46 24)(5 31 19 47 25)(6 32 20 48 26)(7 33 11 49 27)(8 34 12 50 28)(9 35 13 41 29)(10 36 14 42 30)(51 83 65 97 79)(52 84 66 98 80)(53 85 67 99 71)(54 86 68 100 72)(55 87 69 91 73)(56 88 70 92 74)(57 89 61 93 75)(58 90 62 94 76)(59 81 63 95 77)(60 82 64 96 78)
(1 49 39 29 19)(2 50 40 30 20)(3 41 31 21 11)(4 42 32 22 12)(5 43 33 23 13)(6 44 34 24 14)(7 45 35 25 15)(8 46 36 26 16)(9 47 37 27 17)(10 48 38 28 18)(51 91 81 71 61)(52 92 82 72 62)(53 93 83 73 63)(54 94 84 74 64)(55 95 85 75 65)(56 96 86 76 66)(57 97 87 77 67)(58 98 88 78 68)(59 99 89 79 69)(60 100 90 80 70)
(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 56 6 51)(2 55 7 60)(3 54 8 59)(4 53 9 58)(5 52 10 57)(11 64 16 69)(12 63 17 68)(13 62 18 67)(14 61 19 66)(15 70 20 65)(21 74 26 79)(22 73 27 78)(23 72 28 77)(24 71 29 76)(25 80 30 75)(31 84 36 89)(32 83 37 88)(33 82 38 87)(34 81 39 86)(35 90 40 85)(41 94 46 99)(42 93 47 98)(43 92 48 97)(44 91 49 96)(45 100 50 95)

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

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

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

200 conjugacy classes

 class 1 2 4A 4B 5A ··· 5X 5Y ··· 5BV 10A ··· 10X 10Y ··· 10BV 20A ··· 20AV order 1 2 4 4 5 ··· 5 5 ··· 5 10 ··· 10 10 ··· 10 20 ··· 20 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 C4 C5 C10 C20 D5 Dic5 C5×D5 C5×Dic5 kernel Dic5×C52 C52×C10 C53 C5×Dic5 C5×C10 C52 C5×C10 C52 C10 C5 # reps 1 1 2 24 24 48 2 2 48 48

Matrix representation of Dic5×C52 in GL4(𝔽41) generated by

 18 0 0 0 0 18 0 0 0 0 10 0 0 0 0 10
,
 37 0 0 0 0 16 0 0 0 0 18 0 0 0 0 18
,
 40 0 0 0 0 1 0 0 0 0 37 0 0 0 0 10
,
 9 0 0 0 0 40 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,10,0,0,0,0,10],[37,0,0,0,0,16,0,0,0,0,18,0,0,0,0,18],[40,0,0,0,0,1,0,0,0,0,37,0,0,0,0,10],[9,0,0,0,0,40,0,0,0,0,0,1,0,0,1,0] >;

Dic5×C52 in GAP, Magma, Sage, TeX

{\rm Dic}_5\times C_5^2
% in TeX

G:=Group("Dic5xC5^2");
// GroupNames label

G:=SmallGroup(500,37);
// by ID

G=gap.SmallGroup(500,37);
# by ID

G:=PCGroup([5,-2,-5,-5,-2,-5,250,10004]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^5=c^10=1,d^2=c^5,a*b=b*a,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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