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#72227872

order1536
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C22×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22C2×#1020⋊C2
commutator, cosocle, hypocenter#543⋊C3C23
FrattiniC24C22×S4
Fitting#23469D3
prime radical#23469⋊C3C2
hypercenterC2×C4#1020⋊C2
LCSG ▷ #543⋊C3
UCSC1 ◁ C22 ◁ C2×C4

812 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 45 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) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^3,3,2^6|b^2=i,c^2=j,e^2=i,f^2=i,b^a=bj,c^a=cj,c^b=cj,d^b=d^2,e^a=eghj,e^b=fi,e^d=f,f^a=fgj,f^b=ei,f^d=efi,f^e=fi,g^b=gh,g^d=h,g^e=gj,h^d=gh,h^e=hj,h^f=hj

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227873

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