Finite Group Databasebeta

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#72233424

order1536
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C25×C4

Aliases

No aliases for this group.

Splittings

FittingC22×#585 ⋊ D3

Subgroups

subgroupG/H
center, FrattiniC2C2×#87588
commutator, cosocle, hypocenterC22⋊#997C22
FittingC22×#585D3
socleC23C2×#1020
hypercenterC4#87588
LCSG ▷ C22⋊#997
UCSC1 ◁ C2 ◁ C4

7436 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 7436 conjugacy classes of subgroups — too many to draw. Only its 39 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) = 20
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 3 generators on 20 points:

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

Polycyclic presentation:

2^2,3,2^7|b^2=j,b^a=bj,c^a=c^2,d^a=e,d^c=e,e^a=d,e^c=de,f^a=eg,f^c=g,g^a=df,g^c=fg,h^c=i,h^d=hj,h^e=hj,h^g=hj,i^a=hi,i^c=hi,i^e=ij,i^f=ij

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72233726, #72241291, #72241296, …
maximal normal subgroup of#72233726, #72241291, #72241296, …
maximal quotient of#72233431, C2×#72233424, #72267109

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