Finite Group Databasebeta

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#72235289

order1536
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C22×C4

Aliases

No aliases for this group.

Splittings

derivedC22⋊(C42⋊C3) ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, hypocenterC22⋊(C42⋊C3)C2×C4
FrattiniC25C2×S4
Fitting#15258D3
socleC24C2×SL(2,ℤ/4)
prime radical#15258⋊C3C2
cosocle#85647C22
LCSG ▷ C22⋊(C42⋊C3)
UCSC1

838 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 838 conjugacy classes of subgroups — too many to draw. Only its 25 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) = 32

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

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

Polycyclic presentation:

2^3,3,2^6|b^2=c,e^2=j,f^2=i,d^b=d^2,e^a=eghij,e^b=efhi,e^c=ej,e^d=fj,f^a=fgj,f^b=fgi,f^c=fi,f^d=ef,g^b=gi,g^d=h,h^b=ghj,h^d=gh,i^d=ij,j^b=ij,j^d=i

Conjugacy classes & character table

33 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72235302, #72235442

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