Finite Group Databasebeta

← Prev (#72272086) Next (#72272088) →

#72272087

order3072
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22C2×#87524
commutator, cosocle#41⋊SL(2,𝔽3)C23
FrattiniC2×C4C22×#1020
Fitting#41×2−1+4D3
prime radical#72268198C2
hypercenterC4×D4#1020
hypocenterQ8⋊SL(2,𝔽3)C2×D4
LCSG ▷ #41⋊SL(2,𝔽3) ▷ Q8⋊SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

8365 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 8365 conjugacy classes of subgroups — too many to draw. Only its 77 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) = 48
transitive degreeμt(G) = 384

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^4,3,2^6|b^2=jk,c^2=j,d^2=jk,f^2=k,g^2=k,h^2=j,i^2=j,b^a=bd,c^b=cj,d^a=djk,d^b=djk,e^b=e^2,f^b=gk,f^e=g,g^b=fk,g^e=fgk,g^f=gk,h^b=hj,h^e=i,h^f=hj,h^g=hj,i^b=hi,i^e=hij,i^g=ij,i^h=ij

Conjugacy classes & character table

85 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72272131
maximal normal subgroup of#72272131

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