Finite Group Databasebeta

← Prev (#72261989) Next (#72261991) →

#72261990

order3072
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC2C2×#11710955
commutator, hypocenterC22⋊(#95⋊C3)C2×C4
FrattiniC26C2×S4
FittingC2×#7555D3
socleC25C2×SL(2,ℤ/4)
prime radicalC2×#7555⋊C3C2
cosocle#7555⋊C3C22
hypercenterC22#11710955
LCSG ▷ C22⋊(#95⋊C3)
UCSC1 ◁ C2 ◁ C22

4227 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 4227 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) = 24
transitive degreeμt(G) = 24

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

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

Polycyclic presentation:

2^3,3,2^7|b^2=c,e^2=i,f^2=ij,b^a=bij,d^a=dj,d^b=d^2,e^b=efhijk,e^c=eh,e^d=fik,f^b=fghij,f^c=fg,f^d=efk,f^e=fk,g^d=gh,h^b=gh,h^d=g,i^b=jk,i^d=ij,j^b=ik,j^d=ik

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72262094

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