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#72226610

order18432
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting, socleC28 ⋊ GO+(4,𝔽2)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, cosocle#72220401C22
Fitting, socleC28GO+(4,𝔽2)
hypocenterC22⋊A42D4
LCSG ▷ #72220401 ▷ C22⋊A42
UCSC1

At least 6993 conjugacy classes of subgroups; only the 11 normal subgroups are shown.

normal subgroups only

This group has at least 6993 conjugacy classes of subgroups — too many to draw. Only its 11 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 16

A minimal faithful permutation representation — 2 generators on 16 points:

(1,2,4,8,11,7)(3,6,16,12,10,14)(5,15)(9,13);(1,4)(2,6)(3,9)(5,11)(7,13)(8,12)(10,16)(14,15)

Polycyclic presentation:

2^3,3^2,2^8|b^a=bc,d^a=d^2,d^b=e,d^c=d^2,e^b=d,e^c=e^2,f^b=j,f^c=g,f^e=g,g^b=k,g^c=f,g^e=fg,h^b=jl,h^c=fi,h^e=hi,i^b=jklm,i^c=gh,i^e=h,j^a=k,j^b=f,j^c=k,j^d=k,k^a=j,k^b=g,k^c=j,k^d=jk,l^a=lm,l^b=fh,l^c=lm,l^d=m,m^b=ghi,m^d=lm

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72258970
maximal normal subgroup of#72258970
automorphism group ofQ82

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