Finite Group Databasebeta

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#72222921

order10368
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

FittingC22×C34 ⋊ C2×SD16
SylowC34 ⋊ C2×#465

Subgroups

subgroupG/H
center, FrattiniC2C2×#72210346
commutator, cosocleC2×#66472C24
FittingC22×C34C2×SD16
socleC2×C34C22×SD16
prime radical#72220567C2
hypercenterC22#72210346
hypocenterC34C2×#465
LCSG ▷ C2×#66472 ▷ C3⋊(C3⋊(C3⋊D3)) ▷ C34
UCSC1 ◁ C2 ◁ C22

2479 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 2479 conjugacy classes of subgroups — too many to draw. Only its 135 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:

2^7,3^4|c^2=e,e^2=g,b^a=bf,c^a=ce,e^a=eg,h^a=h^2,h^c=i,h^d=h^2,h^e=hi^2,h^g=h^2,i^a=hi,i^c=hi^2,i^d=i^2,i^e=h^2i^2,i^g=i^2,j^a=hi^2j^2,j^c=k,j^d=hij,j^e=jk^2,j^g=j^2,k^a=i^2jk,k^c=jk^2,k^d=hk,k^e=j^2k^2,k^g=k^2

Conjugacy classes & character table

78 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72222932
maximal normal subgroup of#72222932

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