Finite Group Databasebeta

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#72252886

order10368
automorphism?
homologyAbelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C3×C4

Aliases

No aliases for this group.

Splittings

SylowC34 ⋊ C4×M32

Subgroups

subgroupG/H
center, FrattiniC2C2×#157×AGL(1,𝔽9)
commutator, socleC2×C34C2×C4×C8
FittingC22×C34C4×C8
prime radicalC22×C3⋊(C3⋊(C3⋊D3))C2×C8
cosocle#13049399C23
hypercenterC22#157×AGL(1,𝔽9)
hypocenterC34C4×M32
LCSG ▷ C2×C34 ▷ C34
UCSC1 ◁ C2 ◁ C22

916 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 916 conjugacy classes of subgroups — too many to draw. Only its 112 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:

2^7,3^4|a^2=d,c^2=e,d^2=f,f^2=g,b^a=bg,h^a=i,h^c=h^2i,h^d=hi^2,h^e=h^2,h^f=h^2,i^a=hi^2,i^c=hi,i^d=h^2i^2,i^e=i^2,i^f=i^2,j^a=hj,j^c=jk,j^d=hij,j^e=j^2,j^f=hi^2j,k^a=hik,k^c=jk^2,k^d=h^2ik,k^e=k^2,k^f=ik

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72269065
maximal normal subgroup of#72269065

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