Finite Group Databasebeta

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#72342017

order3072
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting#6629390 ⋊ D3

Subgroups

subgroupG/H
center, socleC22C2×#73081
commutator, cosocleC4×#997C23
FrattiniC23×C4C22×S4
Fitting#6629390D3
hypercenterC42#1020⋊C2
hypocenter#997AGL(1,ℤ/8)
LCSG ▷ C4×#997 ▷ C2×#997 ▷ #997
UCSC1 ◁ C22 ◁ C42

2491 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 2491 conjugacy classes of subgroups — too many to draw. Only its 49 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) = 128

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^5,3,2^5|a^2=ek,b^2=k,c^2=ek,d^2=e,g^2=k,h^2=k,b^a=bd,c^b=ce,d^a=de,d^b=de,f^b=f^2,g^a=gik,g^b=gh,g^f=h,h^a=hjk,h^b=hk,h^f=ghk,h^g=hk,i^b=ij,i^f=j,i^h=ik,j^f=ij,j^g=jk

Conjugacy classes & character table

73 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72342092
maximal normal subgroup of#72342092

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