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#72220798

order10368
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

SylowC34 ⋊ #2014

Subgroups

subgroupG/H
center, Frattini, hypercenterC2#72213315
commutator, cosocleC2×#10620198C23
Fitting, socleC2×C34#492
prime radicalC2×#72213247C2
hypocenterC34#2014
LCSG ▷ C2×#10620198 ▷ C3⋊(C3⋊(C3⋊D3)) ▷ C34
UCSC1 ◁ C2

1816 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 1816 conjugacy classes of subgroups — too many to draw. Only its 72 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|a^2=f,b^2=d,d^2=g,b^a=be,c^a=cf,c^b=cdg,d^c=dg,h^a=hi,h^b=i^2,h^c=h^2i^2,h^d=hi,h^f=h^2,h^g=h^2,i^a=hi^2,i^b=h^2i^2,i^d=hi^2,i^f=i^2,i^g=i^2,j^a=h^2j,j^b=k,j^c=j^2,j^d=jk^2,j^f=hi^2j,j^g=j^2,k^a=ik,k^b=jk^2,k^c=jk,k^d=j^2k^2,k^f=hk,k^g=k^2

Conjugacy classes & character table

60 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, 1, F.6);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 4, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.9^2);;
gap> SetConjugate(coll, 8, 3, F.8^2*F.9^2);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 8, 6, F.8^2);;
gap> SetConjugate(coll, 8, 7, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.8*F.9^2);;
gap> SetConjugate(coll, 9, 2, F.8^2*F.9^2);;
gap> SetConjugate(coll, 9, 4, F.8*F.9^2);;
gap> SetConjugate(coll, 9, 6, F.9^2);;
gap> SetConjugate(coll, 9, 7, F.9^2);;
gap> SetConjugate(coll, 10, 1, F.8^2*F.10);;
gap> SetConjugate(coll, 10, 2, F.11);;
gap> SetConjugate(coll, 10, 3, F.10^2);;
gap> SetConjugate(coll, 10, 4, F.10*F.11^2);;
gap> SetConjugate(coll, 10, 6, F.8*F.9^2*F.10);;
gap> SetConjugate(coll, 10, 7, F.10^2);;
gap> SetConjugate(coll, 11, 1, F.9*F.11);;
gap> SetConjugate(coll, 11, 2, F.10*F.11^2);;
gap> SetConjugate(coll, 11, 3, F.10*F.11);;
gap> SetConjugate(coll, 11, 4, F.10^2*F.11^2);;
gap> SetConjugate(coll, 11, 6, F.8*F.11);;
gap> SetConjugate(coll, 11, 7, F.11^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);