Finite Group Databasebeta

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#72269081

order5184
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C3

Aliases

No aliases for this group.

Splittings

SylowC34 ⋊ #423

Subgroups

subgroupG/H
center, FrattiniC2C2×#13049781
commutatorC2×C3⋊(C3⋊(C3⋊D3))C22×C4
FittingC22×C34C4⋊C4
socleC2×C34C2×C4⋊C4
prime radicalC22×C3⋊(C3⋊(C3⋊D3))C2×C4
cosocle#10620150C23
hypercenterC22#13049781
hypocenterC34#423
LCSG ▷ C2×C3⋊(C3⋊(C3⋊D3)) ▷ C34
UCSC1 ◁ C2 ◁ C22

978 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 978 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 48

Polycyclic presentation:

2^6,3^4|a^2=d,b^2=e,c^2=f,d^2=f,b^a=be,c^a=cf,g^a=gh,g^b=gh^2,g^c=g^2,g^d=g^2,g^e=g^2,h^a=gh^2,h^b=g^2h^2,h^c=h^2,h^d=h^2,h^e=h^2,i^a=h^2i^2,i^b=j,i^c=g^2h^2i,i^d=g^2h^2i,i^e=i^2,j^a=g^2h^2j,j^b=i^2,j^c=h^2j,j^d=h^2j,j^e=j^2

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72269114, #72269129
maximal normal subgroup of#72269114, #72269129

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