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#72256006

order3072
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting#8093108 ⋊ D3

Subgroups

subgroupG/H
center, hypercenterC2#72234656
commutator, cosocle#35346⋊C3C22
FrattiniC24C2×#1020
Fitting#8093108D3
socleC23#87524
hypocenterC22⋊#997D4
LCSG ▷ #35346⋊C3 ▷ C22⋊#997
UCSC1 ◁ C2

4727 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 4727 conjugacy classes of subgroups — too many to draw. Only its 30 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:

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

Polycyclic presentation:

2^3,3,2^7|b^2=c,b^a=bc,d^b=d^2,e^b=f,e^d=f,f^b=e,f^d=ef,g^a=gj,g^b=ghij,g^c=gi,g^d=h,g^e=gk,h^a=hi,h^b=hi,h^c=hij,h^d=gh,h^e=hk,h^f=hk,i^b=j,i^d=ij,j^b=i,j^d=i

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72256075

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