Finite Group Databasebeta

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#72235201

order1536
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×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#1437⋊C3C22
LCSG ▷ C22⋊(C42⋊C3)
UCSC1

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

normal subgroups only

This group has 781 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) = 28
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 2 generators on 28 points:

(1,8,4,3)(2,9,7,11,5,6,10,12)(13,22,28,19,16,18,27,23)(14,26,24,17,21,20,15,25);(1,2,9)(3,10,11,8,7,12)(4,5,6)(14,26,15,18,20,23)(16,17,25)(19,22,28,24,21,27)

Polycyclic presentation:

2^3,3,2^6|b^2=c,e^2=j,f^2=i,d^a=dj,d^b=d^2,e^a=eg,e^b=fgi,e^c=egj,e^d=ef,f^a=fh,f^b=egh,f^c=fhi,f^d=ei,g^b=h,g^d=gh,h^b=g,h^d=g,i^b=j,i^d=j,j^b=i,j^d=ij

Conjugacy classes & character table

33 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72235206, #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, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.6*F.7*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.5*F.7*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 4, F.5*F.9);;
gap> SetConjugate(coll, 7, 2, F.8);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 8, 2, F.7);;
gap> SetConjugate(coll, 8, 4, F.7);;
gap> SetConjugate(coll, 9, 2, F.10);;
gap> SetConjugate(coll, 9, 4, F.10);;
gap> SetConjugate(coll, 10, 2, F.9);;
gap> SetConjugate(coll, 10, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);