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G = He5⋊5C4  order 500 = 22·53

1st semidirect product of He5 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial

Aliases: He5⋊5C4, C52⋊2C20, C52⋊2Dic5, (C5×C10).C10, C52⋊6C4⋊C5, C10.2(C5×D5), (C5×C10).1D5, C2.(C52⋊C10), (C2×He5).1C2, C5.2(C5×Dic5), SmallGroup(500,8)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C52 — He5⋊5C4
Chief series
C1 — C5 — C52 — C5×C10 — C2×He5 — He5⋊5C4
Lower central
C52 — He5⋊5C4
Upper central
C1 — C2

Generators and relations for He5⋊5C4
 G = < a,b,c,d | a5=b5=c5=d4=1, cac-1=ab=ba, dad-1=ac3, bc=cb, dbd-1=b-1, dcd-1=c-1 >

C1
C2
C5
5C5
5C5
10C5
10C5
25C4
C10
5C10
5C10
10C10
10C10
C52
C52
2C52
2C52
5Dic5
25Dic5
25C20
C5×C10
C5×C10
2C5×C10
2C5×C10
He5
C52⋊6C4
5C5×Dic5
C2×He5
He5⋊5C4

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

(2 65 37 51 49)(3 38 50 61 52)(4 53 62 46 39)(5 47 54 40 63)(6 30 74 18 83)(7 75 84 26 19)(8 20 27 85 71)(9 81 16 72 28)(11 77 94 21 66)(12 95 67 78 22)(13 23 79 68 91)(14 69 24 92 80)(31 96 90 45 56)(33 58 42 87 98)(34 43 99 59 88)(35 89 60 100 44)

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

44 conjugacy classes

class 1  2 4A4B5A5B5C5D5E5F5G···5P10A10B10C10D10E10F10G···10P20A···20H
order12445555555···510101010101010···1020···20
size11252522555510···1022555510···1025···25

44 irreducible representations

dim11111110102222
type+++-+-
imageC1C2C4C5C10C20C52⋊C10He5⋊5C4D5Dic5C5×D5C5×Dic5
kernelHe5⋊5C4C2×He5He5C52⋊6C4C5×C10C52C2C1C5×C10C52C10C5
# reps112448222288

Matrix representation of He5⋊5C4 ►in GL12(𝔽41)

6400000000000
100000000000
000010000000
000001000000
000000610000
0000004000000
0000000035600
00000000354000
00000000004035
0000000000635
0004000000000
001600000000
,
100000000000
010000000000
006100000000
0040000000000
000061000000
0000400000000
000000610000
0000004000000
000000006100
0000000040000
000000000061
0000000000400
,
6400000000000
100000000000
001000000000
000100000000
000061000000
0000400000000
0000003560000
00000035400000
00000000403500
0000000063500
0000000000040
000000000016
,
900000000000
13320000000000
00133900000000
0022800000000
00001339000000
0000228000000
00000013390000
0000002280000
00000000133900
0000000022800
00000000001339
0000000000228

G:=sub<GL(12,GF(41))| [6,1,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,6,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,6,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,35,35,0,0,0,0,0,0,0,0,0,0,6,40,0,0,0,0,0,0,0,0,0,0,0,0,40,6,0,0,0,0,0,0,0,0,0,0,35,35,0,0],[1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,6,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,6,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,6,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,6,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,6,40,0,0,0,0,0,0,0,0,0,0,1,0],[6,1,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,6,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,35,35,0,0,0,0,0,0,0,0,0,0,6,40,0,0,0,0,0,0,0,0,0,0,0,0,40,6,0,0,0,0,0,0,0,0,0,0,35,35,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,6],[9,13,0,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,0,0,13,2,0,0,0,0,0,0,0,0,0,0,39,28,0,0,0,0,0,0,0,0,0,0,0,0,13,2,0,0,0,0,0,0,0,0,0,0,39,28,0,0,0,0,0,0,0,0,0,0,0,0,13,2,0,0,0,0,0,0,0,0,0,0,39,28,0,0,0,0,0,0,0,0,0,0,0,0,13,2,0,0,0,0,0,0,0,0,0,0,39,28,0,0,0,0,0,0,0,0,0,0,0,0,13,2,0,0,0,0,0,0,0,0,0,0,39,28] >;
 

He5⋊5C4 in GAP, Magma, Sage, TeX 

{\rm He}_5\rtimes_5C_4
 
% in TeX
 
G:=Group("He5:5C4");
 
// GroupNames label
 
G:=SmallGroup(500,8);
 
// by ID
 
G=gap.SmallGroup(500,8);
 
# by ID
 
G:=PCGroup([5,-2,-5,-2,-5,-5,50,1603,1208,10004]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^5=b^5=c^5=d^4=1,c*a*c^-1=a*b=b*a,d*a*d^-1=a*c^3,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of He5⋊5C4 in TeX

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