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## G = C4×5- 1+2order 500 = 22·53

### Direct product of C4 and 5- 1+2

direct product, metacyclic, nilpotent (class 2), monomial, 5-elementary

Aliases: C4×5- 1+2, C100⋊C5, C253C20, C52.C20, C50.2C10, C20.2C52, (C5×C20).C5, C5.2(C5×C20), C10.3(C5×C10), (C5×C10).3C10, C2.(C2×5- 1+2), (C2×5- 1+2).2C2, SmallGroup(500,14)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C4×5- 1+2
 Chief series C1 — C5 — C10 — C5×C10 — C2×5- 1+2 — C4×5- 1+2
 Lower central C1 — C5 — C4×5- 1+2
 Upper central C1 — C20 — C4×5- 1+2

Generators and relations for C4×5- 1+2
G = < a,b,c | a4=b25=c5=1, ab=ba, ac=ca, cbc-1=b6 >

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

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

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

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

116 conjugacy classes

 class 1 2 4A 4B 5A 5B 5C 5D 5E 5F 5G 5H 10A 10B 10C 10D 10E 10F 10G 10H 20A ··· 20H 20I ··· 20P 25A ··· 25T 50A ··· 50T 100A ··· 100AN order 1 2 4 4 5 5 5 5 5 5 5 5 10 10 10 10 10 10 10 10 20 ··· 20 20 ··· 20 25 ··· 25 50 ··· 50 100 ··· 100 size 1 1 1 1 1 1 1 1 5 5 5 5 1 1 1 1 5 5 5 5 1 ··· 1 5 ··· 5 5 ··· 5 5 ··· 5 5 ··· 5

116 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 5 5 5 type + + image C1 C2 C4 C5 C5 C10 C10 C20 C20 5- 1+2 C2×5- 1+2 C4×5- 1+2 kernel C4×5- 1+2 C2×5- 1+2 5- 1+2 C100 C5×C20 C50 C5×C10 C25 C52 C4 C2 C1 # reps 1 1 2 20 4 20 4 40 8 4 4 8

Matrix representation of C4×5- 1+2 in GL5(𝔽101)

 91 0 0 0 0 0 91 0 0 0 0 0 91 0 0 0 0 0 91 0 0 0 0 0 91
,
 95 71 0 22 14 0 0 1 0 0 0 0 0 87 0 0 0 0 0 95 83 17 14 65 6
,
 95 6 5 22 36 0 36 0 0 0 0 0 87 0 0 0 0 0 84 0 0 0 0 0 1

G:=sub<GL(5,GF(101))| [91,0,0,0,0,0,91,0,0,0,0,0,91,0,0,0,0,0,91,0,0,0,0,0,91],[95,0,0,0,83,71,0,0,0,17,0,1,0,0,14,22,0,87,0,65,14,0,0,95,6],[95,0,0,0,0,6,36,0,0,0,5,0,87,0,0,22,0,0,84,0,36,0,0,0,1] >;

C4×5- 1+2 in GAP, Magma, Sage, TeX

C_4\times 5_-^{1+2}
% in TeX

G:=Group("C4xES-(5,1)");
// GroupNames label

G:=SmallGroup(500,14);
// by ID

G=gap.SmallGroup(500,14);
# by ID

G:=PCGroup([5,-2,-5,-5,-2,-5,250,506,3082]);
// Polycyclic

G:=Group<a,b,c|a^4=b^25=c^5=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^6>;
// generators/relations

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