Finite Group Databasebeta

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#72252850

order2592
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2

Aliases

No aliases for this group.

Splittings

Fitting, socle, SylowC34 ⋊ #103

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutatorC3⋊(C3⋊#157)C2×C4
Fitting, socle, hypocenterC34#103
prime radicalC3⋊D32D4
cosocleC3⋊D3×#157C22
LCSG ▷ C3⋊(C3⋊#157) ▷ C32×C3⋊D3 ▷ C34
UCSC1

192 conjugacy classes of subgroups; only the 20 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 20 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) = 48

Polycyclic presentation:

2^5,3^4|a^2=e,b^2=d,c^2=e,b^a=bc,c^a=ce,c^b=ce,f^a=fg,f^b=f^2,f^c=g,f^e=f^2,g^a=fg^2,g^c=f^2,g^e=g^2,h^a=f^2g^2h^2,h^b=hi,h^c=fh^2,h^d=fg^2h^2,h^e=f^2gh,i^a=f^2h^2i,i^b=fg^2hi^2,i^c=f^2g^2i^2,i^d=fi^2,i^e=f^2i

Conjugacy classes & character table

33 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72252851, #72252894, #72252907
maximal normal subgroup of#72252851, #72252894, #72252907

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