Finite Group Databasebeta

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#72258629

order1536
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

FittingC23×#127 ⋊ D3

Subgroups

subgroupG/H
centerC22C2×#857⋊C2
commutator, cosocleC22×C42⋊C3C23
Frattini, socleC24C22×S4
FittingC23×#127D3
hypercenterC23#857⋊C2
hypocenterC42⋊C3#120
LCSG ▷ C22×C42⋊C3 ▷ C42⋊C3
UCSC1 ◁ C22 ◁ C23

3526 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 3526 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 16
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 3 generators on 16 points:

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

Polycyclic presentation:

2^5,3,2^4|b^2=de,g^2=i,h^2=j,b^a=bd,c^b=ceij,f^b=f^2i,f^e=fi,g^b=hij,g^c=gi,g^f=gh,h^b=gij,h^c=hj,h^f=gj,i^b=j,i^f=ij,j^b=i,j^f=i

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72258638
maximal normal subgroup of#72258638
maximal quotient of#72258719

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