Finite Group Databasebeta

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#72258558

order1536
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting, SylowC22×#2032 ⋊ C3
derivedC22×#543 ⋊ C2×C3

Subgroups

subgroupG/H
centerC2#11709844
commutator, cosocle, hypocenterC22×#543C2×C3
FrattiniC24C2×C22⋊A4
FittingC22×#2032C3
socleC23#393⋊C3
hypercenterC22#85645
LCSG ▷ C22×#543
UCSC1 ◁ C2 ◁ C22

1798 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 1798 conjugacy classes of subgroups — too many to draw. Only its 34 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)(5,6,9,12)(7,11,20,17,8,18,14,19,13,10,15,16);(1,4)(2,3);(5,13)(6,16,12,18)(7,8)(9,20)(10,19,17,11)(14,15)

Polycyclic presentation:

2,3,2^8|a^2=j,c^2=i,d^2=i,c^a=cg,c^b=cd,d^a=dgh,d^b=c,d^c=di,e^b=ef,f^b=e,g^b=hj,g^c=gj,g^d=gj,h^b=ghj,h^c=hj,i^a=ij

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72258614, #72258646

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