Finite Group Databasebeta

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#72224521

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

Aliases

No aliases for this group.

Splittings

Fitting, socle, SylowC35 ⋊ SD16

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, cosocleC3×#66472C22
Fitting, socle, hypocenterC35SD16
prime radicalC3×#10620206C2
LCSG ▷ C3×#66472 ▷ C3×C3⋊(C3⋊(C3⋊D3)) ▷ C35
UCSC1

558 conjugacy classes of subgroups; only the 17 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 17 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 2 generators on 15 points:

(2,3)(4,9)(5,12,10,7)(6,8)(11,15,13,14);(1,2,3)(5,8)(7,12)(9,15)(10,14)

Polycyclic presentation:

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

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#13049804, #72224541, #72224565
maximal normal subgroup ofD3×#13049804, #72224541, #72224565

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