Finite Group Databasebeta

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#72261837

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

Aliases

No aliases for this group.

Splittings

FittingC2×#2679 ⋊ D3

Subgroups

subgroupG/H
centerC2C2×#85559
commutator, cosocle, hypocenter#2679⋊C3C22
Frattini, socleC23C2×#1020
FittingC2×#2679D3
hypercenterC22#85559
LCSG ▷ #2679⋊C3
UCSC1 ◁ C2 ◁ C22

1277 conjugacy classes of subgroups; only the 19 normal subgroups are shown.

normal subgroups only

This group has 1277 conjugacy classes of subgroups — too many to draw. Only its 19 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:

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

Polycyclic presentation:

2^2,3,2^7|a^2=hij,b^a=bhij,c^a=c^2,c^b=chi,d^a=de,d^c=e,e^c=de,f^c=g,f^d=fh,f^e=fhj,g^a=fg,g^c=fg,g^d=gi,g^e=gj,h^a=j,h^c=j,i^c=h,j^a=h,j^c=i

Conjugacy classes & character table

31 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72261861

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