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#72272066

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#72209718
commutator, cosocle#11710659C23
FrattiniC42C22×#1020
FittingH(5,𝔽4)D3
prime radical#72268464C2
hypercenterH(3,𝔽4)#1020
hypocenter#5465#120
LCSG ▷ #11710659 ▷ #5465
UCSC1 ◁ C22 ◁ C42 ◁ H(3,𝔽4)

At least 23322 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has at least 23322 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) = 96
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 3 generators on 96 points:

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

Polycyclic presentation:

2^5,3,2^6|c^2=kl,d^2=k,e^2=kl,g^2=k,h^2=k,i^2=kl,j^2=kl,c^a=ce,c^b=cd,d^a=dl,d^b=dk,d^c=dk,e^a=ekl,e^b=el,e^c=ekl,f^c=f^2,g^c=gh,g^f=h,h^c=hk,h^f=ghk,h^g=hk,i^c=ij,i^f=j,i^h=il,j^c=jkl,j^f=ijkl,j^g=jl,j^i=jkl

Conjugacy classes & character table

97 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72272090

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.11*F.12);;
gap> SetPower(coll, 4, F.11);;
gap> SetPower(coll, 5, F.11*F.12);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.11);;
gap> SetPower(coll, 9, F.11*F.12);;
gap> SetPower(coll, 10, F.11*F.12);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.12);;
gap> SetConjugate(coll, 4, 2, F.4*F.11);;
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.12);;
gap> SetConjugate(coll, 5, 3, F.5*F.11*F.12);;
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.9*F.10);;
gap> SetConjugate(coll, 9, 6, F.10);;
gap> SetConjugate(coll, 9, 8, F.9*F.12);;
gap> SetConjugate(coll, 10, 3, F.10*F.11*F.12);;
gap> SetConjugate(coll, 10, 6, F.9*F.10*F.11*F.12);;
gap> SetConjugate(coll, 10, 7, F.10*F.12);;
gap> SetConjugate(coll, 10, 9, F.10*F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);