Finite Group Databasebeta

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#72228620

order3888
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C22×C3

Aliases

No aliases for this group.

Splittings

Fitting, socle, SylowC35 ⋊ C22⋊C4

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutatorC33×C3⋊D3C2×C4
Fitting, socle, hypocenterC35C22⋊C4
prime radicalC3⋊D3×#1113C2
cosocleC3×C3⋊D32C22
LCSG ▷ C33×C3⋊D3 ▷ C35
UCSC1

1132 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 1132 conjugacy classes of subgroups — too many to draw. Only its 28 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:

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

Polycyclic presentation:

2^4,3^5|b^2=c,b^a=bd,e^a=e^2,f^a=fh,f^b=fh,f^c=f^2,f^d=f^2,g^a=gh,g^b=gi,g^c=g^2,g^d=fg,h^a=h^2,h^b=fh^2,h^c=h^2,h^d=h^2,i^a=hi,i^b=gi^2,i^c=i^2,i^d=hi

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

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