Finite Group Databasebeta

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#72210405

order11664
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

No aliases for this group.

Splittings

FittingH(5,𝔽3) ⋊ GL(2,𝔽3)
derived#72209952 ⋊ C2

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenter#72209952C2
Frattini, socleC3#72210292
FittingH(5,𝔽3)GL(2,𝔽3)
LCSG ▷ #72209952
UCSC1

547 conjugacy classes of subgroups; only the 11 normal subgroups are shown.

normal subgroups only

This group has 547 conjugacy classes of subgroups — too many to draw. Only its 11 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,b^a=b^2,c^a=de,c^b=cd,d^a=ce,d^b=c,d^c=de,f^a=f^2,f^c=fg,f^d=gj^2,f^e=f^2j^2,g^a=fg,g^b=fgj,g^c=fg^2j,g^d=f^2,g^e=g^2j,g^f=gj,h^a=fg^2h^2,h^b=fhj,h^c=fhi^2j,h^d=f^2g^2i^2j,h^e=h^2j^2,h^f=hj,i^a=f^2gh^2i,i^b=f^2gh^2i,i^c=f^2gh^2i^2,i^d=fg^2h,i^e=i^2,i^f=ij^2,i^h=ij,j^a=j^2

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72214644, C2×#72210405, #72254100
maximal normal subgroup of#72214644, C2×#72210405, #72254100
maximal quotient ofC2×#72210405
holomorph ofH(3,𝔽3)

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