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#72227778

order6144
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C23×C4

Aliases

No aliases for this group.

Splittings

Fitting#72227752 ⋊ D3

Subgroups

subgroupG/H
center, socleC22#72209653
commutator, cosocle, hypocenter#72227654C22
FrattiniC25C2×#1020
Fitting#72227752D3
hypercenterC23#87588⋊C2
LCSG ▷ #72227654
UCSC1 ◁ C22 ◁ C23

6309 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 6309 conjugacy classes of subgroups — too many to draw. Only its 30 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) = 32
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 2 generators on 32 points:

(1,10,4,3)(2,15,8,9)(5,12,14,11)(6,16,13,7)(17,23,21,32)(18,20)(19,31,28,22)(24,29)(25,27)(26,30);(1,2)(3,13,10,9,4,15)(5,6)(7,14,16,11,8,12)(17,18)(19,25,29,22,26,32)(20,31,27,23,28,30)(21,24)

Polycyclic presentation:

2^2,3,2^9|d^2=l,e^2=l,f^2=j,g^2=j,b^a=bj,c^b=c^2,d^b=deil,d^c=e,e^b=ehi,e^c=del,e^d=el,f^a=fhj,f^b=fgij,f^c=g,f^d=fj,f^e=fk,g^a=ghij,g^b=ghi,g^c=fgj,g^d=gjk,g^e=gj,g^f=gj,h^b=i,h^c=hi,h^e=hl,h^g=hj,i^b=h,i^c=h,i^d=il,i^e=il,i^f=ij,i^g=ij,k^a=kl,k^b=kl

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227800

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