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#72276557

order2592
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C22×C3

Aliases

No aliases for this group.

Splittings

derived(C22⋊A4)⋊C9 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenter(C22⋊A4)⋊C9C2×C3
FrattiniC3C3×A4⋊S4
FittingC24×C32C3⋊D3
socleC24×C3C3×C3⋊D3
LCSG ▷ (C22⋊A4)⋊C9
UCSC1

328 conjugacy classes of subgroups; only the 28 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 28 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) = 35
transitive degreeμt(G) = 54

A minimal faithful permutation representation — 3 generators on 35 points:

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

Polycyclic presentation:

2,3^4,2^4|c^3=e,c^a=c^2e^2,c^b=ce^2,d^a=d^2,d^b=de,e^a=e^2,f^c=g,f^d=fg,g^a=fg,g^c=fg,g^d=f,h^a=fh,h^c=fh,h^d=i,i^a=ghi,i^c=fgi,i^d=hi

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72276569

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