Finite Group Databasebeta

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#72220249

order5184
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C2×C4

Aliases

No aliases for this group.

Splittings

SylowC34 ⋊ #486

Subgroups

subgroupG/H
center, Frattini, hypercenterC2#72210346
commutator, cosocleC2×#66472C23
Fitting, socleC2×C34C2×SD16
prime radical#72219698C2
hypocenterC34#486
LCSG ▷ C2×#66472 ▷ C3⋊(C3⋊(C3⋊D3)) ▷ C34
UCSC1 ◁ C2

682 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 37 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:

2^6,3^4|a^2=d,c^2=ef,d^2=f,b^a=bde,c^b=cef,d^b=df,g^a=h,g^b=g^2h,g^c=g^2,g^d=gh^2,g^e=g^2,g^f=g^2,h^a=gh^2,h^c=h^2,h^d=g^2h^2,h^e=h^2,h^f=h^2,i^a=h^2j,i^b=i^2,i^c=g^2h^2i,i^d=hij^2,i^e=i^2,i^f=i^2,j^a=gij^2,j^b=ij,j^c=g^2hj,j^d=g^2i^2j^2,j^e=j^2,j^f=j^2

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72220288, #72220297, #72220298, …
maximal normal subgroup of#72220288, #72220297, #72220298, …

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