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## G = C5×C25⋊C4order 500 = 22·53

### Direct product of C5 and C25⋊C4

Aliases: C5×C25⋊C4, C254C20, C52.2F5, D25.2C10, (C5×C25)⋊3C4, C5.1(C5×F5), (C5×D25).1C2, SmallGroup(500,16)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C25 — C5×C25⋊C4
 Chief series C1 — C5 — C25 — D25 — C5×D25 — C5×C25⋊C4
 Lower central C25 — C5×C25⋊C4
 Upper central C1 — C5

Generators and relations for C5×C25⋊C4
G = < a,b,c | a5=b25=c4=1, ab=ba, ac=ca, cbc-1=b18 >

Smallest permutation representation of C5×C25⋊C4
On 100 points
Generators in S100
(1 6 11 16 21)(2 7 12 17 22)(3 8 13 18 23)(4 9 14 19 24)(5 10 15 20 25)(26 46 41 36 31)(27 47 42 37 32)(28 48 43 38 33)(29 49 44 39 34)(30 50 45 40 35)(51 66 56 71 61)(52 67 57 72 62)(53 68 58 73 63)(54 69 59 74 64)(55 70 60 75 65)(76 86 96 81 91)(77 87 97 82 92)(78 88 98 83 93)(79 89 99 84 94)(80 90 100 85 95)
(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 83 50 60)(2 90 49 53)(3 97 48 71)(4 79 47 64)(5 86 46 57)(6 93 45 75)(7 100 44 68)(8 82 43 61)(9 89 42 54)(10 96 41 72)(11 78 40 65)(12 85 39 58)(13 92 38 51)(14 99 37 69)(15 81 36 62)(16 88 35 55)(17 95 34 73)(18 77 33 66)(19 84 32 59)(20 91 31 52)(21 98 30 70)(22 80 29 63)(23 87 28 56)(24 94 27 74)(25 76 26 67)

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

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

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

50 conjugacy classes

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

50 irreducible representations

 dim 1 1 1 1 1 1 4 4 4 4 type + + + + image C1 C2 C4 C5 C10 C20 F5 C25⋊C4 C5×F5 C5×C25⋊C4 kernel C5×C25⋊C4 C5×D25 C5×C25 C25⋊C4 D25 C25 C52 C5 C5 C1 # reps 1 1 2 4 4 8 1 5 4 20

Matrix representation of C5×C25⋊C4 in GL4(𝔽101) generated by

 36 0 0 0 0 36 0 0 0 0 36 0 0 0 0 36
,
 25 0 0 0 0 97 0 0 0 0 78 0 0 0 0 79
,
 0 0 1 0 0 0 0 1 0 1 0 0 1 0 0 0
G:=sub<GL(4,GF(101))| [36,0,0,0,0,36,0,0,0,0,36,0,0,0,0,36],[25,0,0,0,0,97,0,0,0,0,78,0,0,0,0,79],[0,0,0,1,0,0,1,0,1,0,0,0,0,1,0,0] >;

C5×C25⋊C4 in GAP, Magma, Sage, TeX

C_5\times C_{25}\rtimes C_4
% in TeX

G:=Group("C5xC25:C4");
// GroupNames label

G:=SmallGroup(500,16);
// by ID

G=gap.SmallGroup(500,16);
# by ID

G:=PCGroup([5,-2,-5,-2,-5,-5,50,2803,973,118,5004,1014]);
// Polycyclic

G:=Group<a,b,c|a^5=b^25=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^18>;
// generators/relations

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