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#72267768

order6144
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C2×C4

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72267764 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC2#72226619
commutator, Fitting, cosocle, hypocenter#72267764C3
FrattiniC27C22⋊A4
socleC23#11709340
LCSG ▷ #72267764
UCSC1 ◁ C2

4302 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 4302 conjugacy classes of subgroups — too many to draw. Only its 44 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) = 40
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 3 generators on 40 points:

(2,5,3)(6,7,8)(10,16,26)(11,20,32)(12,13,15)(14,29,28)(17,37,25)(18,36,24)(19,35,22)(21,34,23)(30,33,31)(38,39,40);(2,6)(5,7)(10,30,18,38)(11,26)(14,27)(16,33,32,37)(17,21,19,20)(22,25)(23,24)(28,29)(31,40)(34,35,36,39);(1,7,4,5)(2,3,6,8)(9,32,13,34)(10,23,18,26)(11,21,24,20)(12,16,15,36)(14,39,28,33)(17,31,19,22)(25,30,40,38)(27,35,29,37)

Polycyclic presentation:

3,2^11|b^2=f,c^2=fgh,d^2=hjl,e^2=ghjl,b^a=cgkl,c^a=bcfg,c^b=cfg,d^a=ehk,d^b=dfi,d^c=dhj,e^a=dejkl,e^b=efgij,e^c=efhi,e^d=ejkl,f^a=fghl,f^c=fk,g^a=hk,g^b=gk,g^c=gl,h^a=ghl,h^b=hl,h^c=hkl,i^a=gik,i^c=il,i^d=ikl,i^e=il,j^a=jk,j^b=jk,j^c=jkl,k^a=kl,l^a=k

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72267769
maximal normal subgroup of#72267769

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