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#72268883

order3072
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC22#11711503
commutatorC22×#95⋊C3C2×C4
FrattiniC24×C4C2×S4
FittingC2×#15172D3
socleC24#5432
prime radicalC2×#15172⋊C3C2
cosocleC22×#392⋊C3C22
hypercenterC22×C4#4626
hypocenter#95⋊C3#99
LCSG ▷ C22×#95⋊C3 ▷ C2×#95⋊C3 ▷ #95⋊C3
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4

2343 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 2343 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 96

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

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

Polycyclic presentation:

2^5,3,2^5|a^2=c,b^2=ek,c^2=k,g^2=i,h^2=j,b^a=bd,c^b=ce,d^a=de,f^a=f^2ik,f^b=fik,f^c=fik,g^a=gh,g^c=gik,g^f=hik,h^a=hijk,h^c=hjk,h^f=ghk,h^g=hk,i^a=ijk,i^f=j,j^f=ijk

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72268885
maximal normal subgroup of#72268885

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