Finite Group Databasebeta

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#72248138

order3072
automorphism?
homologyAbelianization (H1) = C42 · Schur multiplier (H2) = C22×C4

Aliases

No aliases for this group.

Splittings

derivedC22⋊(C22⋊A4) ⋊ C42

Subgroups

subgroupG/H
center, hypercenterC2#72215879
commutator, hypocenterC22⋊(C22⋊A4)C42
FrattiniC26C2×S4
FittingC2×#8541D3
socleC23#1020⋊C4
prime radical, cosocleC2×#85645C22
LCSG ▷ C22⋊(C22⋊A4)
UCSC1 ◁ C2

3339 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 3339 conjugacy classes of subgroups — too many to draw. Only its 38 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 — 2 generators on 20 points:

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

Polycyclic presentation:

2^4,3,2^6|a^2=d,b^2=c,e^b=e^2,f^a=fh,f^d=fk,f^e=g,g^a=gi,g^b=fg,g^d=gj,g^e=fg,h^a=hk,h^e=i,i^a=ij,i^b=hi,i^e=hi,j^b=jk,j^e=jk,k^e=j

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72248174, #72253665, #72253798, …
maximal normal subgroup of#72248174, #72253665, #72253798, …

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.11);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.6*F.7);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.8*F.9);;
gap> SetConjugate(coll, 9, 5, F.8*F.9);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> SetConjugate(coll, 10, 5, F.10*F.11);;
gap> SetConjugate(coll, 11, 5, F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);