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#72268175

order6144
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72268170 ⋊ C3
derived#72268147 ⋊ C2×C3

Subgroups

subgroupG/H
center, socleC23#11710337
commutator, cosocle, hypocenter#72268147C2×C3
FrattiniC23×Q8C2×C22⋊A4
Fitting#72268170C3
hypercenterC24#85645
LCSG ▷ #72268147
UCSC1 ◁ C23 ◁ C24

6336 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 6336 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 256

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2,3,2^10|c^2=j,d^2=j,e^2=k,f^2=k,g^2=l,h^2=l,c^a=cg,c^b=cd,d^a=dh,d^b=c,d^c=dj,e^b=ef,e^c=ei,e^d=eil,f^b=e,f^c=fl,f^d=fi,f^e=fk,g^a=gl,g^b=ghijl,g^c=gl,g^d=gijl,g^e=gl,h^a=hl,h^b=g,h^c=hij,h^d=hl,h^e=hl,h^f=hl,h^g=hl,i^a=il

Conjugacy classes & character table

99 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72268201
maximal normal subgroup of#72268201

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