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#72227397

order3072
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72219804 ⋊ C3
derivedC22×#1304 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC22×#1304C2×C3
FrattiniC27C2×A4
Fitting#72219804C3
socleC24C2×#95⋊C3
LCSG ▷ C22×#1304
UCSC1

37931 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 37931 conjugacy classes of subgroups — too many to draw. Only its 31 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) = 48

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

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

Polycyclic presentation:

2,3,2^9|c^2=g,d^2=gh,c^a=cef,c^b=dgijk,d^a=de,d^b=cdi,d^c=di,e^b=f,f^b=ef,g^b=ghk,g^d=gj,h^b=gijk,h^c=hjk,h^d=hj,i^b=ijk,i^c=ij,i^d=ijk,j^b=jk,k^b=j

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72227398

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