Finite Group Databasebeta

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#72224531

order3888
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting, socle, SylowC35 ⋊ C2×D4

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, cosocleC3×C3⋊(C3⋊(C3⋊D3))C23
Fitting, socle, hypocenterC35C2×D4
LCSG ▷ C3×C3⋊(C3⋊(C3⋊D3)) ▷ C35
UCSC1

1304 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 1304 conjugacy classes of subgroups — too many to draw. Only its 33 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 — 3 generators on 15 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

63 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72224533, #72224541, D3×#13049813
maximal normal subgroup of#72224533, #72224541, D3×#13049813

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