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## G = F5×C52order 500 = 22·53

### Direct product of C52 and F5

Aliases: F5×C52, C531C4, C524C20, C5⋊(C5×C20), D5.(C5×C10), (C5×D5).2C10, (D5×C52).1C2, SmallGroup(500,41)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — F5×C52
 Chief series C1 — C5 — D5 — C5×D5 — D5×C52 — F5×C52
 Lower central C5 — F5×C52
 Upper central C1 — C52

Generators and relations for F5×C52
G = < a,b,c,d | a5=b5=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 160 in 60 conjugacy classes, 32 normal (8 characteristic)
C1, C2, C4, C5, C5, C5, D5, C10, C20, F5, C52, C52, C52, C5×D5, C5×C10, C5×C20, C5×F5, C53, D5×C52, F5×C52
Quotients: C1, C2, C4, C5, C10, C20, F5, C52, C5×C10, C5×C20, C5×F5, F5×C52

Smallest permutation representation of F5×C52
On 100 points
Generators in S100
(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 6 73 55 37)(2 7 74 51 38)(3 8 75 52 39)(4 9 71 53 40)(5 10 72 54 36)(11 87 69 91 34)(12 88 70 92 35)(13 89 66 93 31)(14 90 67 94 32)(15 86 68 95 33)(16 85 62 49 26)(17 81 63 50 27)(18 82 64 46 28)(19 83 65 47 29)(20 84 61 48 30)(21 97 79 56 44)(22 98 80 57 45)(23 99 76 58 41)(24 100 77 59 42)(25 96 78 60 43)
(1 38 52 71 10)(2 39 53 72 6)(3 40 54 73 7)(4 36 55 74 8)(5 37 51 75 9)(11 93 86 35 67)(12 94 87 31 68)(13 95 88 32 69)(14 91 89 33 70)(15 92 90 34 66)(16 84 65 46 27)(17 85 61 47 28)(18 81 62 48 29)(19 82 63 49 30)(20 83 64 50 26)(21 77 45 96 58)(22 78 41 97 59)(23 79 42 98 60)(24 80 43 99 56)(25 76 44 100 57)
(1 78 28 69)(2 79 29 70)(3 80 30 66)(4 76 26 67)(5 77 27 68)(6 60 18 91)(7 56 19 92)(8 57 20 93)(9 58 16 94)(10 59 17 95)(11 55 25 64)(12 51 21 65)(13 52 22 61)(14 53 23 62)(15 54 24 63)(31 75 45 84)(32 71 41 85)(33 72 42 81)(34 73 43 82)(35 74 44 83)(36 100 50 86)(37 96 46 87)(38 97 47 88)(39 98 48 89)(40 99 49 90)

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

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

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

125 conjugacy classes

 class 1 2 4A 4B 5A ··· 5X 5Y ··· 5AW 10A ··· 10X 20A ··· 20AV order 1 2 4 4 5 ··· 5 5 ··· 5 10 ··· 10 20 ··· 20 size 1 5 5 5 1 ··· 1 4 ··· 4 5 ··· 5 5 ··· 5

125 irreducible representations

 dim 1 1 1 1 1 1 4 4 type + + + image C1 C2 C4 C5 C10 C20 F5 C5×F5 kernel F5×C52 D5×C52 C53 C5×F5 C5×D5 C52 C52 C5 # reps 1 1 2 24 24 48 1 24

Matrix representation of F5×C52 in GL5(𝔽41)

 37 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 16
,
 18 0 0 0 0 0 10 0 0 0 0 0 10 0 0 0 0 0 10 0 0 0 0 0 10
,
 1 0 0 0 0 0 18 0 0 0 0 0 16 0 0 0 0 0 37 0 0 0 0 0 10
,
 40 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 1 0 0 0

G:=sub<GL(5,GF(41))| [37,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[18,0,0,0,0,0,10,0,0,0,0,0,10,0,0,0,0,0,10,0,0,0,0,0,10],[1,0,0,0,0,0,18,0,0,0,0,0,16,0,0,0,0,0,37,0,0,0,0,0,10],[40,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,1,0,0,0,0,0,1,0,0] >;

F5×C52 in GAP, Magma, Sage, TeX

F_5\times C_5^2
% in TeX

G:=Group("F5xC5^2");
// GroupNames label

G:=SmallGroup(500,41);
// by ID

G=gap.SmallGroup(500,41);
# by ID

G:=PCGroup([5,-2,-5,-5,-2,-5,250,5004,219]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^5=c^5=d^4=1,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^3>;
// generators/relations

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