Finite Group Databasebeta

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#72214508

order11664
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C3

Aliases

No aliases for this group.

Splittings

FittingH(5,𝔽3) ⋊ C2×SL(2,𝔽3)
derived#14519166 ⋊ C2×C3

Subgroups

subgroupG/H
center, Frattini, socle, hypercenterC3#72210295
commutator, cosocle, hypocenter#14519166C2×C3
FittingH(5,𝔽3)C2×SL(2,𝔽3)
LCSG ▷ #14519166
UCSC1 ◁ C3

363 conjugacy classes of subgroups; only the 15 normal subgroups are shown.

normal subgroups only

This group has 363 conjugacy classes of subgroups — too many to draw. Only its 15 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:

2,3,2^3,3^5|c^2=e,d^2=e,c^b=d,d^b=cde,d^c=de,f^a=f^2j^2,f^c=fh,f^d=f^2hj,f^e=f^2j^2,g^b=h^2ij^2,g^c=gh^2i,g^d=h^2ij^2,g^e=g^2j^2,h^a=h^2j,h^b=fhj,h^c=fh^2j,h^d=fhj,h^e=h^2j,h^f=hj,i^a=hij,i^b=fg^2h^2i^2j^2,i^c=fgi^2j^2,i^d=fg^2hj,i^e=i^2j^2,i^f=ij,i^g=ij

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72214644
maximal normal subgroup of#72214644

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