Finite Group Databasebeta

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#72240363

order6144
automorphism?
homologyAbelianization (H1) = C4 · Schur multiplier (H2) = C22×C4

Aliases

No aliases for this group.

Splittings

derived#72240347 ⋊ C4

Subgroups

subgroupG/H
center, hypercenterC2#72240362
commutator, hypocenter#72240347C4
FrattiniC26#1020
Fitting#72233925D3
socleC25#5936
prime radical, cosocle#72240348C2
LCSG ▷ #72240347
UCSC1 ◁ C2

5422 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 5422 conjugacy classes of subgroups — too many to draw. Only its 40 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) = 36
transitive degreeμt(G) = 48

A minimal faithful permutation representation — 2 generators on 36 points:

(1,7,2,8,4,6,5,3)(9,16,11,12)(10,19,14,18,15,13,20,17)(21,33,36,28,22,29,35,25)(23,24,34,30,26,27,32,31);(2,3,7)(5,8,6)(9,10,13)(11,15,19)(12,14,17)(16,20,18)(22,23,26)(24,28,35)(25,32,33)(27,31,34)(29,30,36)

Polycyclic presentation:

2^2,3,2^9|a^2=b,d^2=h,e^2=i,f^2=hkl,g^2=ik,c^a=c^2,d^a=dh,d^c=ehj,e^a=de,e^c=dej,e^d=ej,f^a=deghil,f^b=fk,f^c=ghjkl,f^d=fl,f^e=fjk,g^a=efj,g^b=gl,g^c=fgj,g^d=gjk,g^e=gkl,g^f=gj,h^c=i,i^a=hij,i^c=hij,k^a=l,k^c=l,l^a=k,l^c=kl

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72261698, #72261895, #72262094
maximal normal subgroup of#72261698, #72261895, #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", "l");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.2);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.8*F.11*F.12);;
gap> SetPower(coll, 7, F.9*F.11);;
gap> SetConjugate(coll, 3, 1, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 3, F.4*F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.4*F.5*F.7*F.8*F.9*F.12);;
gap> SetConjugate(coll, 6, 2, F.6*F.11);;
gap> SetConjugate(coll, 6, 3, F.7*F.8*F.10*F.11*F.12);;
gap> SetConjugate(coll, 6, 4, F.6*F.12);;
gap> SetConjugate(coll, 6, 5, F.6*F.10*F.11);;
gap> SetConjugate(coll, 7, 1, F.5*F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.12);;
gap> SetConjugate(coll, 7, 3, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10*F.11);;
gap> SetConjugate(coll, 7, 5, F.7*F.11*F.12);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.9);;
gap> SetConjugate(coll, 9, 1, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 11, 1, F.12);;
gap> SetConjugate(coll, 11, 3, F.12);;
gap> SetConjugate(coll, 12, 1, F.11);;
gap> SetConjugate(coll, 12, 3, F.11*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);