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#72227992

order3072
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C2×C4

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72227991 ⋊ C3
derived#475912 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC2×C4#3549⋊C3
commutator, cosocle, hypocenter#475912C2×C3
FrattiniC23×C4C2×C22⋊A4
Fitting#72227991C3
socleC22#60045⋊C3
LCSG ▷ #475912
UCSC1 ◁ C2×C4

1292 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 1292 conjugacy classes of subgroups — too many to draw. Only its 41 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 — 3 generators on 24 points:

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

Polycyclic presentation:

2,3,2^9|c^2=g,d^2=g,e^2=hj,f^2=ik,h^2=k,j^2=k,c^a=ci,c^b=d,d^a=dhij,d^b=cdg,d^c=dg,e^b=fhj,e^c=eghk,e^d=egjk,f^b=ef,f^c=fi,f^d=fghik,h^b=hik,h^c=hk,i^b=hij,i^d=ik,j^b=hk,j^d=jk

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227995
maximal normal subgroup of#72227995
maximal quotient of#72268645

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