Finite Group Databasebeta

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#72252442

order3072
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Splittings

Fitting#10574065 ⋊ D3

Subgroups

subgroupG/H
centerC4D4×#1020
commutator#5950C22×C4
FrattiniC2×C4C22×#1020
Fitting#10574065D3
socleC2C22⋊C4×#1020
prime radical#72215767C2
cosocleC2×#5950C23
hypercenter#104#1020
hypocenter#997C2×C22⋊C4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C4 ◁ C2×C4 ◁ #104

At least 6549 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has at least 6549 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

2^5,3,2^5|a^2=e,b^2=e,c^2=k,d^2=k,g^2=k,h^2=k,c^a=cd,c^b=cd,d^c=dk,e^c=ek,f^b=f^2,g^b=gh,g^f=h,h^b=hk,h^f=ghk,h^g=hk,i^b=j,i^f=j,i^g=ik,i^h=ik,j^b=i,j^f=ij,j^h=jk

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72252511
maximal normal subgroup of#72252511

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