Finite Group Databasebeta

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#72219701

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

Aliases

No aliases for this group.

Splittings

SylowC34 ⋊ #103

Subgroups

subgroupG/H
center, Frattini, hypercenterC2#13049803
commutator#66472C2×C4
Fitting, socleC2×C34SD16
prime radicalC2×C3⋊(C3⋊(C3⋊D3))D4
cosocleC2×#66472C22
hypocenterC34#103
LCSG ▷ #66472 ▷ C3⋊(C3⋊(C3⋊D3)) ▷ C34
UCSC1 ◁ C2

383 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 383 conjugacy classes of subgroups — too many to draw. Only its 24 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:

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

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72219704, #72219720, #72219725, …
maximal normal subgroup of#72219704, #72219720, #72219725, …

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