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#72245682

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

Aliases

No aliases for this group.

Splittings

Fitting#41×3−1+2 ⋊ D3

Subgroups

subgroupG/H
center, hypercenterC4A4⋊AGL(1,ℤ/9)
commutator, cosocle, hypocenterC9×SL(2,𝔽3)C22×C3
Frattini, socleC2×C3C2×C3×C3⋊S4
Fitting#41×3−1+2D3
LCSG ▷ C9×SL(2,𝔽3)
UCSC1 ◁ C4

388 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 53 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) = 25
transitive degreeμt(G) = 144

A minimal faithful permutation representation — 3 generators on 25 points:

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

Polycyclic presentation:

2^2,3^4,2^3|a^2=i,e^3=f,g^2=i,h^2=i,d^b=d^2,e^b=e^2f^2,e^c=ef,f^b=f^2,g^b=hi,g^d=gh,g^e=gh,h^b=gi,h^d=g,h^e=g,h^g=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72245689, #72253122
maximal normal subgroup of#72253122

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