Finite Group Databasebeta

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#72258970

order36864
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

Fitting#6342371 ⋊ GO+(4,𝔽2)

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle#72220401C23
Frattini, socleC24C2×#10596770⋊C2
Fitting#6342371GO+(4,𝔽2)
hypocenterC22⋊A42C2×D4
LCSG ▷ #72220401 ▷ C22⋊A42
UCSC1

At least 22920 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has at least 22920 conjugacy classes of subgroups — too many to draw. Only its 24 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:

2^4,3^2,2^8|c^2=d,c^b=cd,e^b=e^2,e^c=e^2f^2,e^d=e^2,f^b=e^2f,f^c=e^2f,f^d=f^2,g^a=gkl,g^b=j,g^c=jk,g^d=gh,g^e=h,g^f=hn,h^a=hk,h^b=ij,h^c=ij,h^e=gh,h^f=ghm,i^a=im,i^b=gh,i^c=gmn,i^d=j,i^e=ij,i^f=il,j^a=jn,j^b=g,j^c=ghmn,j^d=i,j^e=i,j^f=jk,k^b=mn,k^c=mn,k^e=l,k^f=l,l^b=m,l^c=m,l^d=kl,l^e=kl,l^f=kl,m^b=l,m^c=kl,m^d=n,m^e=mn,n^b=kl,n^c=l,n^d=m,n^e=m

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

automorphism group ofC2×H(5,𝔽2)

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", "l", "m", "n");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 5, 3, F.5^2*F.6^2);;
gap> SetConjugate(coll, 5, 4, F.5^2);;
gap> SetConjugate(coll, 6, 2, F.5^2*F.6);;
gap> SetConjugate(coll, 6, 3, F.5^2*F.6);;
gap> SetConjugate(coll, 6, 4, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7*F.11*F.12);;
gap> SetConjugate(coll, 7, 2, F.10);;
gap> SetConjugate(coll, 7, 3, F.10*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 7, 5, F.8);;
gap> SetConjugate(coll, 7, 6, F.8*F.14);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> SetConjugate(coll, 8, 2, F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.9*F.10);;
gap> SetConjugate(coll, 8, 5, F.7*F.8);;
gap> SetConjugate(coll, 8, 6, F.7*F.8*F.13);;
gap> SetConjugate(coll, 9, 1, F.9*F.13);;
gap> SetConjugate(coll, 9, 2, F.7*F.8);;
gap> SetConjugate(coll, 9, 3, F.7*F.13*F.14);;
gap> SetConjugate(coll, 9, 4, F.10);;
gap> SetConjugate(coll, 9, 5, F.9*F.10);;
gap> SetConjugate(coll, 9, 6, F.9*F.12);;
gap> SetConjugate(coll, 10, 1, F.10*F.14);;
gap> SetConjugate(coll, 10, 2, F.7);;
gap> SetConjugate(coll, 10, 3, F.7*F.8*F.13*F.14);;
gap> SetConjugate(coll, 10, 4, F.9);;
gap> SetConjugate(coll, 10, 5, F.9);;
gap> SetConjugate(coll, 10, 6, F.10*F.11);;
gap> SetConjugate(coll, 11, 2, F.13*F.14);;
gap> SetConjugate(coll, 11, 3, F.13*F.14);;
gap> SetConjugate(coll, 11, 5, F.12);;
gap> SetConjugate(coll, 11, 6, F.12);;
gap> SetConjugate(coll, 12, 2, F.13);;
gap> SetConjugate(coll, 12, 3, F.13);;
gap> SetConjugate(coll, 12, 4, F.11*F.12);;
gap> SetConjugate(coll, 12, 5, F.11*F.12);;
gap> SetConjugate(coll, 12, 6, F.11*F.12);;
gap> SetConjugate(coll, 13, 2, F.12);;
gap> SetConjugate(coll, 13, 3, F.11*F.12);;
gap> SetConjugate(coll, 13, 4, F.14);;
gap> SetConjugate(coll, 13, 5, F.13*F.14);;
gap> SetConjugate(coll, 14, 2, F.11*F.12);;
gap> SetConjugate(coll, 14, 3, F.12);;
gap> SetConjugate(coll, 14, 4, F.13);;
gap> SetConjugate(coll, 14, 5, F.13);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);