Finite Group Databasebeta

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#72271504

order3072
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting, SylowQ8×#2202 ⋊ C3
derivedQ8×H(3,𝔽4) ⋊ C2×C3

Subgroups

subgroupG/H
center, socleC22#11710337
commutator, cosocle, hypocenterQ8×H(3,𝔽4)C2×C3
FrattiniC22×Q8C2×C22⋊A4
FittingQ8×#2202C3
hypercenterC23#85645
LCSG ▷ Q8×H(3,𝔽4)
UCSC1 ◁ C22 ◁ C23

1995 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 1995 conjugacy classes of subgroups — too many to draw. Only its 36 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) = 128

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2,3,2^9|a^2=k,e^2=j,f^2=j,g^2=k,h^2=k,c^a=cg,c^b=d,d^a=dh,d^b=cd,e^b=f,f^b=efj,f^e=fj,g^a=gk,g^b=h,g^c=gk,g^d=gi,h^a=hk,h^b=ghi,h^c=hik,h^d=hk,h^g=hk,i^a=ik

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72271548

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