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#72227995

order6144
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72227994 ⋊ C3
derived#475912 ⋊ C3×C4

Subgroups

subgroupG/H
center, hypercenterC2×C4#8530⋊C3
commutator, hypocenter#475912C3×C4
FrattiniC22×C43C2×A4
Fitting#72227994C3
socleC22#72209726
prime radical#72227992C2
cosocle#72227991C2×C3
LCSG ▷ #475912
UCSC1 ◁ C2×C4

2180 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 2180 conjugacy classes of subgroups — too many to draw. Only its 38 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) = 24
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 2 generators on 24 points:

(2,8,3,5,6,7)(10,23,22,15,20,19,17,24,11,18,16,21);(1,8,4,6)(2,3,5,7)(9,20,14,16)(10,11,17,22)(12,23,13,24)(15,21,18,19)

Polycyclic presentation:

2,3,2^10|a^2=c,d^2=h,e^2=h,f^2=ikl,g^2=j,i^2=l,k^2=l,b^a=bjl,d^a=dg,d^b=e,d^c=dj,e^a=ef,e^b=deh,e^c=eikl,e^d=eh,f^b=fghi,f^d=fhil,f^e=fikl,g^b=fl,g^d=gj,g^e=ghil,i^b=kl,i^d=il,i^e=il,j^b=ikl,j^e=jl,k^b=ij,k^e=kl

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227996
maximal quotient of#72268646

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