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#72227462

order6144
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C2×C4

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72227460 ⋊ C3
derived#72210018 ⋊ C2×C3

Subgroups

subgroupG/H
center, socle, hypercenterC2#72227461
commutator, cosocle, hypocenter#72210018C2×C3
FrattiniC2×#2867C2×A4
Fitting#72227460C3
LCSG ▷ #72210018
UCSC1 ◁ C2

1653 conjugacy classes of subgroups; only the 19 normal subgroups are shown.

normal subgroups only

This group has 1653 conjugacy classes of subgroups — too many to draw. Only its 19 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) = 16
transitive degreeμt(G) = 16

A minimal faithful permutation representation — 2 generators on 16 points:

(2,12,11,8,3,15,9,10,16,6,13,7);(1,7,4,11)(2,12,6,3)(5,15,14,16)(8,13,9,10)

Polycyclic presentation:

2,3,2^10|a^2=ij,c^2=g,d^2=g,e^2=j,f^2=k,b^a=bjl,c^a=ce,c^b=cd,d^a=defk,d^b=c,d^c=dg,e^b=fghjkl,e^c=ei,e^d=egh,f^b=efghkl,f^c=fghjk,f^d=fghik,g^a=gij,g^e=gj,g^f=gk,h^a=hij,h^b=hjl,h^c=hk,h^d=hjl,h^e=hj,h^f=hk,h^g=hl,i^b=ijkl,i^c=ij,i^d=ijkl,j^b=kl,j^d=jl,k^b=jkl,k^c=kl,k^d=kl

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227463
maximal quotient of#72268817

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