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## G = C5×C5⋊F5order 500 = 22·53

### Direct product of C5 and C5⋊F5

Aliases: C5×C5⋊F5, C533C4, C525F5, C526C20, C51(C5×F5), C5⋊D5.2C10, (C5×C5⋊D5).1C2, SmallGroup(500,43)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C5×C5⋊F5
G = < a,b,c,d | a5=b5=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=c3 >

Subgroups: 288 in 60 conjugacy classes, 20 normal (8 characteristic)
C1, C2, C4, C5, C5, C5, D5, C10, C20, F5, C52, C52, C52, C5×D5, C5⋊D5, C5×F5, C5⋊F5, C53, C5×C5⋊D5, C5×C5⋊F5
Quotients: C1, C2, C4, C5, C10, C20, F5, C5×F5, C5⋊F5, C5×C5⋊F5

Smallest permutation representation of C5×C5⋊F5
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 34 45 37 26)(2 35 41 38 27)(3 31 42 39 28)(4 32 43 40 29)(5 33 44 36 30)(6 97 25 17 14)(7 98 21 18 15)(8 99 22 19 11)(9 100 23 20 12)(10 96 24 16 13)(46 57 68 54 65)(47 58 69 55 61)(48 59 70 51 62)(49 60 66 52 63)(50 56 67 53 64)(71 79 82 90 93)(72 80 83 86 94)(73 76 84 87 95)(74 77 85 88 91)(75 78 81 89 92)
(1 40 35 30 42)(2 36 31 26 43)(3 37 32 27 44)(4 38 33 28 45)(5 39 34 29 41)(6 16 100 11 21)(7 17 96 12 22)(8 18 97 13 23)(9 19 98 14 24)(10 20 99 15 25)(46 51 56 61 66)(47 52 57 62 67)(48 53 58 63 68)(49 54 59 64 69)(50 55 60 65 70)(71 86 76 91 81)(72 87 77 92 82)(73 88 78 93 83)(74 89 79 94 84)(75 90 80 95 85)
(1 21 70 95)(2 22 66 91)(3 23 67 92)(4 24 68 93)(5 25 69 94)(6 55 80 30)(7 51 76 26)(8 52 77 27)(9 53 78 28)(10 54 79 29)(11 60 85 35)(12 56 81 31)(13 57 82 32)(14 58 83 33)(15 59 84 34)(16 65 90 40)(17 61 86 36)(18 62 87 37)(19 63 88 38)(20 64 89 39)(41 99 49 74)(42 100 50 75)(43 96 46 71)(44 97 47 72)(45 98 48 73)

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

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

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

50 conjugacy classes

 class 1 2 4A 4B 5A 5B 5C 5D 5E ··· 5AH 10A 10B 10C 10D 20A ··· 20H order 1 2 4 4 5 5 5 5 5 ··· 5 10 10 10 10 20 ··· 20 size 1 25 25 25 1 1 1 1 4 ··· 4 25 25 25 25 25 ··· 25

50 irreducible representations

 dim 1 1 1 1 1 1 4 4 type + + + image C1 C2 C4 C5 C10 C20 F5 C5×F5 kernel C5×C5⋊F5 C5×C5⋊D5 C53 C5⋊F5 C5⋊D5 C52 C52 C5 # reps 1 1 2 4 4 8 6 24

Matrix representation of C5×C5⋊F5 in GL8(𝔽41)

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

G:=sub<GL(8,GF(41))| [37,0,0,0,0,0,0,0,0,37,0,0,0,0,0,0,0,0,37,0,0,0,0,0,0,0,0,37,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[18,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,37,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,37,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,37,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,37],[0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;

C5×C5⋊F5 in GAP, Magma, Sage, TeX

C_5\times C_5\rtimes F_5
% in TeX

G:=Group("C5xC5:F5");
// GroupNames label

G:=SmallGroup(500,43);
// by ID

G=gap.SmallGroup(500,43);
# by ID

G:=PCGroup([5,-2,-5,-2,-5,-5,50,803,173,5004,1014]);
// 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,d*b*d^-1=b^3,d*c*d^-1=c^3>;
// generators/relations

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