Finite Group Databasebeta

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#72252865

order5184
automorphism?
homologyAbelianization (H1) = C4×C16 · Schur multiplier (H2) = C3×C4

Aliases

No aliases for this group.

Splittings

derived, SylowC34 ⋊ C4×C16

Subgroups

subgroupG/H
center, Frattini, hypercenterC2#157×AGL(1,𝔽9)
commutator, hypocenterC34C4×C16
Fitting, socleC2×C34C4×C8
prime radicalC2×C3⋊(C3⋊(C3⋊D3))C2×C8
cosocle#13049399C22
LCSG ▷ C34
UCSC1 ◁ C2

276 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 40 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) = 96

Polycyclic presentation:

2^6,3^4|a^2=c,b^2=d,c^2=e,e^2=f,g^a=g^2h^2,g^b=gh^2,g^c=g^2h,g^d=g^2,g^e=g^2,h^a=g^2,h^b=g^2h^2,h^c=gh,h^d=h^2,h^e=h^2,i^a=g^2hi,i^b=ij,i^c=g^2h^2i,i^d=i^2,i^e=g^2i,j^a=g^2h^2j,j^b=ij^2,j^c=gj,j^d=j^2,j^e=hj

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72252886, #72269049, #72269051
maximal normal subgroup of#72252886, #72269049, #72269051

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