Finite Group Databasebeta

← Prev (#72241399) Next (#72241401) →

#72241400

order6144
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72241399 ⋊ C3
derived#8934620 ⋊ C3×C4

Subgroups

subgroupG/H
center, socleC2#72216927
commutator, hypocenter#8934620C3×C4
FrattiniC2×H(7,𝔽2)C2×A4
Fitting#72241399C3
prime radical#72227585C2
cosocle#72213228C2×C3
hypercenterC23#8541⋊C3
LCSG ▷ #8934620
UCSC1 ◁ C2 ◁ C22 ◁ C23

3076 conjugacy classes of subgroups; only the 18 normal subgroups are shown.

normal subgroups only

This group has 3076 conjugacy classes of subgroups — too many to draw. Only its 18 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) = 32
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

2,3,2^10|a^2=c,d^2=k,e^2=k,f^2=l,g^2=l,d^a=df,d^b=e,d^c=dh,e^a=eg,e^b=dek,e^c=ehi,e^d=ek,f^a=fhl,f^b=g,f^c=fl,f^e=fj,g^a=ghil,g^b=fgjl,g^c=gl,g^d=gj,g^f=gl,h^a=hl,h^b=hi,h^e=hk,h^g=hl,i^b=hk,i^d=ik,i^e=ik,i^f=il,i^g=il,j^a=jk,j^c=jl,k^a=kl

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72252407, #72252411, #72267055
maximal normal subgroup of#72252407, #72252411, #72267055

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