Finite Group Databasebeta

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#72260120

order15552
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C2×C4

Aliases

No aliases for this group.

Splittings

Fitting, socle, SylowC35 ⋊ #492

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, cosocleC3×#10620198C23
Fitting, socle, hypocenterC35#492
prime radicalC3×#72213247C2
LCSG ▷ C3×#10620198 ▷ C3×C3⋊(C3⋊(C3⋊D3)) ▷ C35
UCSC1

1204 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 1204 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.

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

Full subgroup data →

Presentation

transitive degreeμt(G) = 96

Polycyclic presentation:

2^6,3^5|a^2=e,c^2=d,d^2=f,b^a=be,c^b=cd,d^b=df,g^c=g^2,h^a=hi,h^b=h^2i^2,h^c=i^2,h^d=hi,h^e=h^2,h^f=h^2,i^a=hi^2,i^c=h^2i^2,i^d=hi^2,i^e=i^2,i^f=i^2,j^a=hj,j^b=j^2,j^c=k,j^d=jk^2,j^e=h^2ij,j^f=j^2,k^a=i^2k,k^b=jk,k^c=jk^2,k^d=j^2k^2,k^e=h^2k,k^f=k^2

Conjugacy classes & character table

78 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", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 4, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 7, 3, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8^2*F.9^2);;
gap> SetConjugate(coll, 8, 3, F.9^2);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.8^2);;
gap> SetConjugate(coll, 8, 6, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.8*F.9^2);;
gap> SetConjugate(coll, 9, 3, F.8^2*F.9^2);;
gap> SetConjugate(coll, 9, 4, F.8*F.9^2);;
gap> SetConjugate(coll, 9, 5, F.9^2);;
gap> SetConjugate(coll, 9, 6, F.9^2);;
gap> SetConjugate(coll, 10, 1, F.8*F.10);;
gap> SetConjugate(coll, 10, 2, F.10^2);;
gap> SetConjugate(coll, 10, 3, F.11);;
gap> SetConjugate(coll, 10, 4, F.10*F.11^2);;
gap> SetConjugate(coll, 10, 5, F.8^2*F.9*F.10);;
gap> SetConjugate(coll, 10, 6, F.10^2);;
gap> SetConjugate(coll, 11, 1, F.9^2*F.11);;
gap> SetConjugate(coll, 11, 2, F.10*F.11);;
gap> SetConjugate(coll, 11, 3, F.10*F.11^2);;
gap> SetConjugate(coll, 11, 4, F.10^2*F.11^2);;
gap> SetConjugate(coll, 11, 5, F.8^2*F.11);;
gap> SetConjugate(coll, 11, 6, F.11^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);