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G = He5:6C4  order 500 = 22·53

2nd semidirect product of He5 and C4 acting via C4/C2=C2

non-abelian, supersoluble, monomial

Aliases: He5:6C4, C52:3Dic5, (C5xC10).3D5, C10.4(C5:D5), C2.(He5:C2), (C2xHe5).2C2, C5.2(C52:6C4), SmallGroup(500,11)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C5 — He5 — He5:6C4
Chief series
C1 — C5 — C52 — He5 — C2xHe5 — He5:6C4
Lower central
He5 — He5:6C4
Upper central
C1 — C10

Generators and relations for He5:6C4
 G = < a,b,c,d | a5=b5=c5=d4=1, cac-1=ab=ba, dad-1=a-1b-1, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 189 in 45 conjugacy classes, 19 normal (7 characteristic)
Quotients: C1, C2, C4, D5, Dic5, C5:D5, C52:6C4, He5:C2, He5:6C4
C1
C2
C5
5C5
5C5
5C5
5C5
5C5
5C5
25C4
C10
5C10
5C10
5C10
5C10
5C10
5C10
C52
C52
C52
C52
C52
C52
5Dic5
5Dic5
5Dic5
5Dic5
5Dic5
5Dic5
25C20
C5xC10
C5xC10
C5xC10
C5xC10
C5xC10
C5xC10
He5
5C5xDic5
5C5xDic5
5C5xDic5
5C5xDic5
5C5xDic5
5C5xDic5
C2xHe5
He5:6C4

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

(2 39 26 42 33)(3 27 34 40 43)(4 44 36 35 28)(5 31 45 29 37)(6 25 12 18 99)(7 100 19 13 21)(8 14 96 22 20)(10 17 24 98 11)(46 60 68 55 62)(47 69 63 56 51)(48 52 57 64 70)(49 65 53 66 58)(71 80 82 93 90)(73 87 95 84 77)(74 91 78 88 85)(75 81 89 79 92)

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

44 conjugacy classes

class 1  2 4A4B5A5B5C5D5E···5P10A10B10C10D10E···10P20A···20H
order124455555···51010101010···1020···20
size112525111110···10111110···1025···25

44 irreducible representations

dim1112255
type+++-
imageC1C2C4D5Dic5He5:C2He5:6C4
kernelHe5:6C4C2xHe5He5C5xC10C52C2C1
# reps112121288

Matrix representation of He5:6C4 ►in GL7(F41)

04000000
1600000
0001000
00001800
00000370
00000010
00160000
,
1000000
0100000
00180000
00018000
00001800
00000180
00000018
,
35600000
354000000
0010000
00018000
00003700
00000100
00000016
,
13900000
132800000
00400000
00000040
00000400
00004000
00040000

G:=sub<GL(7,GF(41))| [0,1,0,0,0,0,0,40,6,0,0,0,0,0,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,37,0,0,0,0,0,0,0,10,0],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,18],[35,35,0,0,0,0,0,6,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,37,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,16],[13,13,0,0,0,0,0,9,28,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0] >;
 

He5:6C4 in GAP, Magma, Sage, TeX 

{\rm He}_5\rtimes_6C_4
 
% in TeX
 
G:=Group("He5:6C4");
 
// GroupNames label
 
G:=SmallGroup(500,11);
 
// by ID
 
G=gap.SmallGroup(500,11);
 
# by ID
 
G:=PCGroup([5,-2,-2,-5,-5,-5,10,242,1603,613]);
 
// 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^-1*b^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of He5:6C4 in TeX

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