Finite Group Databasebeta

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#72227705

order3072
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72209710 ⋊ C3
derived#8093108 ⋊ C2×C3

Subgroups

subgroupG/H
center, socle, hypercenterC23#85645
commutator, cosocle, hypocenter#8093108C2×C3
FrattiniC25C2×C22⋊A4
Fitting#72209710C3
LCSG ▷ #8093108
UCSC1 ◁ C23

39045 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 39045 conjugacy classes of subgroups — too many to draw. Only its 54 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:

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

Polycyclic presentation:

2,3,2^9|e^2=j,f^2=j,c^b=cd,d^b=c,e^a=eh,e^b=f,e^c=ek,e^d=ej,f^a=fg,f^b=efj,f^c=fj,f^d=fjk,f^e=fj,g^b=ghi,g^e=gi,h^b=g,h^f=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227742
maximal normal subgroup of#72227742

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