Finite Group Databasebeta

← Prev (#72269048) Next (#72269050) →

#72269049

order10368
automorphism?
homologyAbelianization (H1) = C8×C16 · Schur multiplier (H2) = C8

Aliases

No aliases for this group.

Splittings

derived, SylowC34 ⋊ C8×C16

Subgroups

subgroupG/H
center, Frattini, hypercenterC2AGL(1,𝔽9)2
commutator, hypocenterC34C8×C16
Fitting, socleC2×C34C82
prime radicalC2×C3⋊(C3⋊(C3⋊D3))C4×C8
cosocle#72220201C22
LCSG ▷ C34
UCSC1 ◁ C2

319 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 319 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 144

Polycyclic presentation:

2^7,3^4|a^2=d,b^2=c,c^2=e,d^2=f,f^2=g,h^a=i,h^b=hi,h^c=h^2i,h^d=hi^2,h^e=h^2,h^f=h^2,i^a=hi^2,i^b=h,i^c=hi,i^d=h^2i^2,i^e=i^2,i^f=i^2,j^a=hi^2j,j^b=jk,j^c=j^2k,j^d=ij,j^e=j^2,j^f=h^2j,k^a=h^2i^2k,k^b=j,k^c=jk,k^d=hi^2k,k^e=k^2,k^f=i^2k

Conjugacy classes & character table

162 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, 2, F.3);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 8, 1, F.9);;
gap> SetConjugate(coll, 8, 2, F.8*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, 2, F.8);;
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.9^2*F.10);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> SetConjugate(coll, 10, 3, F.10^2*F.11);;
gap> SetConjugate(coll, 10, 4, F.9*F.10);;
gap> SetConjugate(coll, 10, 5, F.10^2);;
gap> SetConjugate(coll, 10, 6, F.8^2*F.10);;
gap> SetConjugate(coll, 11, 1, F.8^2*F.9^2*F.11);;
gap> SetConjugate(coll, 11, 2, F.10);;
gap> SetConjugate(coll, 11, 3, F.10*F.11);;
gap> SetConjugate(coll, 11, 4, F.8*F.9^2*F.11);;
gap> SetConjugate(coll, 11, 5, F.11^2);;
gap> SetConjugate(coll, 11, 6, F.9^2*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);