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#72269004

order6144
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

FittingH(5,𝔽4) ⋊ D3

Subgroups

subgroupG/H
center, socleC22#72209718
commutator, cosocle#11710659C23
FrattiniC42C22×#1020
FittingH(5,𝔽4)D3
hypercenterH(3,𝔽4)#1020
hypocenter#5465#120
LCSG ▷ #11710659 ▷ #5465
UCSC1 ◁ C22 ◁ C42 ◁ H(3,𝔽4)

At least 8571 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has at least 8571 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 64
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

(1,17,5,4,6,15,18,16)(2,32,24,21,9,25,11,19)(3,30,14,13,12,29,28,27)(7,31,22,10,8,26,20,23)(33,50,36,45,41,51,48,44)(34,39,35,55,42,38,47,54)(37,61,63,59,58,49,53,43)(40,62,64,60,57,52,56,46);(3,12)(4,5,17)(7,8)(10,11,26)(13,28,30,27,14,29)(15,16,18)(19,22,32,21,20,25)(23,24,31)(33,34)(35,48)(36,47)(37,44,57,43,38,60)(39,46,58,45,40,59)(41,42)(49,55,62,63,50,56)(51,64,61,54,52,53);(1,2)(3,7)(4,10)(5,11)(6,9)(8,12)(13,19)(14,20)(15,31)(16,23)(17,26)(18,24)(21,27)(22,28)(25,29)(30,32)(35,47)(36,48)(49,61)(50,51)(52,62)(53,63)(54,55)(56,64)

Polycyclic presentation:

2^5,3,2^6|c^2=ek,d^2=l,e^2=k,g^2=k,h^2=k,c^a=cd,c^b=ce,d^a=dl,d^b=dkl,d^c=dk,e^a=ekl,e^b=ek,f^c=f^2,g^c=gh,g^f=h,h^c=hk,h^f=ghk,h^g=hk,i^c=ghijkl,i^f=j,i^g=ik,i^h=ikl,j^c=hjl,j^f=ij,j^g=jl,j^h=jk

Conjugacy classes & character table

97 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72269016

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.5*F.11);;
gap> SetPower(coll, 4, F.12);;
gap> SetPower(coll, 5, F.11);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.12);;
gap> SetConjugate(coll, 4, 2, F.4*F.11*F.12);;
gap> SetConjugate(coll, 4, 3, F.4*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.11*F.12);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 6, 3, F.6^2);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 3, F.8*F.11);;
gap> SetConjugate(coll, 8, 6, F.7*F.8*F.11);;
gap> SetConjugate(coll, 8, 7, F.8*F.11);;
gap> SetConjugate(coll, 9, 3, F.7*F.8*F.9*F.10*F.11*F.12);;
gap> SetConjugate(coll, 9, 6, F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.11);;
gap> SetConjugate(coll, 9, 8, F.9*F.11*F.12);;
gap> SetConjugate(coll, 10, 3, F.8*F.10*F.12);;
gap> SetConjugate(coll, 10, 6, F.9*F.10);;
gap> SetConjugate(coll, 10, 7, F.10*F.12);;
gap> SetConjugate(coll, 10, 8, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);