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#72268182

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

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72268180 ⋊ C3
derived#72268147 ⋊ C2×C3

Subgroups

subgroupG/H
center, socleC22#72209717
commutator, cosocle, hypocenter#72268147C2×C3
FrattiniC23×Q8C2×C22⋊A4
Fitting#72268180C3
hypercenterC24#85645
LCSG ▷ #72268147
UCSC1 ◁ C22 ◁ C24

5942 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 5942 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 128

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

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

Polycyclic presentation:

2,3,2^10|c^2=j,d^2=j,e^2=l,f^2=l,g^2=k,h^2=k,c^a=ch,c^b=cd,d^a=dg,d^b=c,d^c=dj,e^b=f,e^c=ei,e^d=eil,f^b=efl,f^c=fil,f^d=fl,f^e=fl,g^a=gk,g^b=h,g^c=gij,g^d=gl,g^e=gl,h^a=hk,h^b=ghijkl,h^c=hl,h^d=hijl,h^e=hl,h^f=hl,h^g=hk,i^a=il,j^a=jkl

Conjugacy classes & character table

78 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.12);;
gap> SetPower(coll, 6, F.12);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.3);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, 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*F.6*F.12);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.12);;
gap> SetConjugate(coll, 6, 4, F.6*F.12);;
gap> SetConjugate(coll, 6, 5, F.6*F.12);;
gap> SetConjugate(coll, 7, 1, F.7*F.11);;
gap> SetConjugate(coll, 7, 2, F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.12);;
gap> SetConjugate(coll, 7, 5, F.7*F.12);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> SetConjugate(coll, 8, 2, F.7*F.8*F.9*F.10*F.11*F.12);;
gap> SetConjugate(coll, 8, 3, F.8*F.12);;
gap> SetConjugate(coll, 8, 4, F.8*F.9*F.10*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.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.12);;
gap> SetConjugate(coll, 10, 1, F.10*F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);