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#72226631

order1536
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, hypercenterC2#11715690
commutator, cosocle, hypocenterC22⋊(#95⋊C3)C22
FrattiniC23C2×#1020
FittingC22×#392D3
socleC25C2×S4
prime radical#11709839C2
LCSG ▷ C22⋊(#95⋊C3)
UCSC1 ◁ C2

1964 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 1964 conjugacy classes of subgroups — too many to draw. Only its 23 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) = 24

A minimal faithful permutation representation — 2 generators on 24 points:

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

Polycyclic presentation:

2^2,3,2^7|a^2=j,b^2=j,d^2=h,e^2=hi,b^a=bj,c^a=c^2,c^b=chij,d^a=eh,d^b=dhj,d^c=ehj,e^a=dhi,e^b=ehij,e^c=dej,e^d=ej,f^a=fghj,f^c=g,g^a=gi,g^c=fg,h^a=hi,h^c=hi,i^c=hj

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72226632, #72234221
maximal normal subgroup of#72226632, #72234221
maximal quotient of#72231239, #72238097

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