Finite Group Databasebeta

← Prev (#72228074) Next (#72228076) →

#72228075

order1536
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C4

Aliases

No aliases for this group.

Splittings

Fitting#13378 ⋊ D3

Subgroups

subgroupG/H
center, socleC22#73026
commutator, cosocle, hypocenter#73291C22
FrattiniC22×Q8C2×S4
Fitting#13378D3
hypercenterC23#1020⋊C2
LCSG ▷ #73291
UCSC1 ◁ C22 ◁ C23

423 conjugacy classes of subgroups; only the 18 normal subgroups are shown.

normal subgroups only

This group has 423 conjugacy classes of subgroups — too many to draw. Only its 18 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) = 32

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

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

Polycyclic presentation:

2^2,3,2^7|b^2=h,d^2=h,e^2=h,f^2=j,g^2=j,b^a=bh,c^b=c^2,d^a=dfhij,d^b=de,d^c=e,e^a=eghij,e^b=eh,e^c=deh,e^d=eh,f^b=fg,f^c=g,f^d=fj,f^e=fhi,g^b=gj,g^c=fgj,g^d=ghij,g^e=gj,g^f=gj,i^a=ij,i^b=ij

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72228081

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