Finite Group Databasebeta

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#72239930

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC2C2×#72209990
commutator, prime radical, cosocle, hypocenter#72222569C22
Frattini, socleC25C2×#1020
Fitting#72239227D3
hypercenterC4#72209990
LCSG ▷ #72222569
UCSC1 ◁ C2 ◁ C4

6579 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 6579 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) = 56
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 2 generators on 56 points:

(1,3,4,9)(2,12,7,10,6,11,8,5)(14,23)(18,20)(19,21)(22,24)(25,39,40,36,30,37,43,42)(26,47,48,45,34,46,51,50)(27,29,53,41,38,44,54,28)(31,33,55,49,35,52,56,32);(1,2,5,4,6,10)(3,8,11,9,7,12)(13,14,18,17,22,19)(15,20,24,16,21,23)(25,26,30,34)(27,35,38,31)(28,33,40,49,29,51,41,52,43,32,44,48)(36,47,53,45,37,55,42,46,54,50,39,56)

Polycyclic presentation:

2^2,3,2^9|a^2=l,b^2=l,d^2=h,e^2=i,f^2=j,g^2=k,b^a=bkl,c^a=cjk,c^b=c^2,d^b=eh,d^c=ehl,e^b=di,e^c=del,e^d=el,f^b=fghl,f^c=gjk,f^d=fl,g^b=ghik,g^c=fgj,g^d=gl,g^e=gl,h^b=i,h^c=i,i^b=h,i^c=hil,j^b=jk,j^c=k,k^c=jk

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72239932

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.12);;
gap> SetPower(coll, 2, F.12);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.11*F.12);;
gap> SetConjugate(coll, 3, 1, F.3*F.10*F.11);;
gap> SetConjugate(coll, 3, 2, F.3^2);;
gap> SetConjugate(coll, 4, 2, F.5*F.8);;
gap> SetConjugate(coll, 4, 3, F.5*F.8*F.12);;
gap> SetConjugate(coll, 5, 2, F.4*F.9);;
gap> SetConjugate(coll, 5, 3, F.4*F.5*F.12);;
gap> SetConjugate(coll, 5, 4, F.5*F.12);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8*F.12);;
gap> SetConjugate(coll, 6, 3, F.7*F.10*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.12);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.9*F.11);;
gap> SetConjugate(coll, 7, 3, F.6*F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.12);;
gap> SetConjugate(coll, 7, 5, F.7*F.12);;
gap> SetConjugate(coll, 8, 2, F.9);;
gap> SetConjugate(coll, 8, 3, F.9);;
gap> SetConjugate(coll, 9, 2, F.8);;
gap> SetConjugate(coll, 9, 3, F.8*F.9*F.12);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> SetConjugate(coll, 10, 3, F.11);;
gap> SetConjugate(coll, 11, 3, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);