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#72266371

order6144
automorphism?
homologyAbelianization (H1) = C8 · Schur multiplier (H2) = C43

Aliases

No aliases for this group.

Splittings

derived#11709197 ⋊ C8

Subgroups

subgroupG/H
center, hypercenterC2#72210020
commutator, hypocenter#11709197C8
FrattiniC24×C4#1020
Fitting#72266358D3
socleC25#5936
prime radicalC2×#11709197C4
cosocle#72266361C2
LCSG ▷ #11709197
UCSC1 ◁ C2

4811 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 4811 conjugacy classes of subgroups — too many to draw. Only its 39 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) = 32
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 32 points:

(1,6,5,2)(3,18,15,19)(4,8)(7,24,23,11)(9,12,20,10)(13,14)(16,22,17,21)(25,26,27,29,28,30,31,32);(1,3,11)(2,7,9)(4,15,24)(5,10,21)(6,16,18)(8,17,19)(12,22,14)(13,23,20)

Polycyclic presentation:

2^3,3,2^8|a^2=b,b^2=c,e^2=i,f^2=j,g^2=k,h^2=l,d^a=d^2,e^a=f,e^b=ei,e^d=fi,f^a=ei,f^b=fj,f^d=ef,g^a=ehijl,g^b=gk,g^d=gh,h^a=fgij,h^b=hl,h^d=gl,i^a=j,i^d=j,j^a=i,j^d=ij,k^a=il,k^d=kl,l^a=jk,l^d=k

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72266375

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