Finite Group Databasebeta

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#72252974

order2304
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

FittingC32×C22⋊C4 ⋊ SD16
SylowC32 ⋊ #10565

Subgroups

subgroupG/H
centerC2#11794082
commutator, cosocleC23×#157C23
FrattiniC23C2×AΓL(1,𝔽9)
FittingC32×C22⋊C4SD16
socleC2×C32#2055
prime radicalC2×#64242C22
hypercenterC22⋊C4AΓL(1,𝔽9)
hypocenterC32#10565
LCSG ▷ C23×#157 ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C2 ◁ C23 ◁ C22⋊C4

637 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 637 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.

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

Full subgroup data →

Presentation

transitive degreeμt(G) = 72

Polycyclic presentation:

2^8,3^2|a^2=h,b^2=gh,c^2=fh,d^2=h,e^2=h,f^2=h,b^a=bf,c^a=ce,c^b=cd,d^a=dg,d^b=dh,e^a=eh,e^b=eg,f^a=fh,f^b=fh,i^a=ij,i^c=ij,i^d=j^2,i^e=j^2,i^f=j,i^g=i^2,i^h=i^2,j^a=ij^2,j^b=j^2,j^c=i^2j,j^d=i,j^e=i,j^f=i^2,j^g=j^2,j^h=j^2

Conjugacy classes & character table

51 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72252979, #72253254, #72253260
maximal normal subgroup of#72252979, #72253254, #72253260

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