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G = C4×He5order 500 = 22·53

Direct product of C4 and He5

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

Aliases: C4×He5, C523C20, C20.1C52, (C5×C20)⋊C5, C2.(C2×He5), C5.1(C5×C20), C10.2(C5×C10), (C5×C10).2C10, (C2×He5).3C2, SmallGroup(500,13)

Series: Derived Chief Lower central Upper central

C1C5 — C4×He5
C1C5C10C5×C10C2×He5 — C4×He5
C1C5 — C4×He5
C1C20 — C4×He5

Generators and relations for C4×He5
 G = < a,b,c,d | a4=b5=c5=d5=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >

5C5
5C5
5C5
5C5
5C5
5C5
5C10
5C10
5C10
5C10
5C10
5C10
5C20
5C20
5C20
5C20
5C20
5C20

Smallest permutation representation of C4×He5
On 100 points
Generators in S100
(1 21 62 13)(2 22 63 14)(3 23 64 15)(4 24 65 11)(5 25 61 12)(6 77 46 59)(7 78 47 60)(8 79 48 56)(9 80 49 57)(10 76 50 58)(16 68 27 91)(17 69 28 92)(18 70 29 93)(19 66 30 94)(20 67 26 95)(31 73 42 85)(32 74 43 81)(33 75 44 82)(34 71 45 83)(35 72 41 84)(36 97 54 86)(37 98 55 87)(38 99 51 88)(39 100 52 89)(40 96 53 90)
(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 71 49 36 29)(2 72 50 37 30)(3 73 46 38 26)(4 74 47 39 27)(5 75 48 40 28)(6 51 20 64 85)(7 52 16 65 81)(8 53 17 61 82)(9 54 18 62 83)(10 55 19 63 84)(11 32 78 89 68)(12 33 79 90 69)(13 34 80 86 70)(14 35 76 87 66)(15 31 77 88 67)(21 45 57 97 93)(22 41 58 98 94)(23 42 59 99 95)(24 43 60 100 91)(25 44 56 96 92)
(1 2 26 47 28)(3 39 5 71 72)(4 48 36 37 46)(6 65 8 54 55)(7 17 62 63 20)(9 10 85 16 82)(11 79 86 87 77)(12 34 35 15 89)(13 14 67 78 69)(18 19 51 81 53)(21 22 95 60 92)(23 100 25 45 41)(24 56 97 98 59)(27 75 49 50 73)(29 30 38 74 40)(31 68 33 80 76)(32 90 70 66 88)(42 91 44 57 58)(43 96 93 94 99)(52 61 83 84 64)

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

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

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

116 conjugacy classes

class 1  2 4A4B5A5B5C5D5E···5AB10A10B10C10D10E···10AB20A···20H20I···20BD
order124455555···51010101010···1020···2020···20
size111111115···511115···51···15···5

116 irreducible representations

dim111111555
type++
imageC1C2C4C5C10C20He5C2×He5C4×He5
kernelC4×He5C2×He5He5C5×C20C5×C10C52C4C2C1
# reps112242448448

Matrix representation of C4×He5 in GL6(𝔽41)

3200000
0400000
0040000
0004000
0000400
0000040
,
1600000
038103097
040113234
0000160
0000037
09010283
,
100000
0370000
0037000
0003700
0000370
0000037
,
1800000
0381000
040000
000010
000001
09010283

G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[16,0,0,0,0,0,0,38,4,0,0,9,0,10,0,0,0,0,0,30,11,0,0,10,0,9,32,16,0,28,0,7,34,0,37,3],[1,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,37,0,0,0,0,0,0,37],[18,0,0,0,0,0,0,38,4,0,0,9,0,1,0,0,0,0,0,0,0,0,0,10,0,0,0,1,0,28,0,0,0,0,1,3] >;

C4×He5 in GAP, Magma, Sage, TeX

C_4\times {\rm He}_5
% in TeX

G:=Group("C4xHe5");
// GroupNames label

G:=SmallGroup(500,13);
// by ID

G=gap.SmallGroup(500,13);
# by ID

G:=PCGroup([5,-2,-5,-5,-2,-5,250,832]);
// Polycyclic

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

Export

Subgroup lattice of C4×He5 in TeX

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