Finite Group Databasebeta

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#72214527

order3888
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C3

Aliases

No aliases for this group.

Splittings

Fitting, SylowH(5,𝔽3) ⋊ SD16

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle#11758052C22
Frattini, socleC3#13049804
Fitting, hypocenterH(5,𝔽3)SD16
prime radical#14519140C2
LCSG ▷ #11758052 ▷ #92404 ▷ H(5,𝔽3)
UCSC1

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

normal subgroups only

This group has 242 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:

2^4,3^5|b^2=d,c^2=d,b^a=bc,c^a=cd,c^b=cd,e^a=e^2,e^b=eg,e^c=e^2gi,e^d=e^2i^2,f^a=fg^2hi^2,f^b=fgh^2,f^c=fg^2h,f^d=f^2i,g^a=eg,g^b=eg^2i,g^c=egi,g^d=g^2i,g^e=gi,h^a=eg^2h^2,h^b=ef^2h^2,h^c=efg^2h^2i,h^d=h^2i^2,h^e=hi,h^f=hi,i^a=i^2

Conjugacy classes & character table

23 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72214540, #72214604, C2×#72214527
maximal normal subgroup of#72214604, C2×#72214527
maximal quotient ofC2×#72214527

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