Finite Group Databasebeta

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#72227804

order1536
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#355894 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC2×C4C22⋊(C42⋊C3)
commutator, Fitting, cosocle, hypocenter#355894C3
FrattiniC23×C4C22⋊A4
socleC22C22⋊(#95⋊C3)
LCSG ▷ #355894
UCSC1 ◁ C2×C4

1089 conjugacy classes of subgroups; only the 31 normal subgroups are shown.

normal subgroups only

This group has 1089 conjugacy classes of subgroups — too many to draw. Only its 31 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) = 24
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 3 generators on 24 points:

(2,5,3)(6,7,8)(10,16,22)(11,19,12)(17,18,20)(21,23,24);(2,6)(3,8)(9,21,13,22,15,17,14,11)(10,24,23,12,19,20,18,16);(1,7,4,5)(2,3,6,8)(9,24,14,12,15,20,13,16)(10,22,18,17,19,11,23,21)

Polycyclic presentation:

3,2^9|b^2=f,c^2=g,d^2=gi,e^2=fghi,f^2=j,g^2=j,h^2=j,b^a=cfhj,c^a=bch,c^b=ch,d^a=egh,d^b=dhi,d^c=di,e^a=dehij,e^b=ehj,e^c=ehij,e^d=ehij,f^a=gj,f^c=fj,f^d=fj,f^e=fj,g^a=fghj,g^b=gj,g^e=gj

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72227806, #72227818
maximal subgroup of#72227806, #72227809, #72267247, …
maximal normal subgroup of#72227806, #72227809, #72267334

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