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#72231982

order6144
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

FittingC2×#8932885 ⋊ D3

Subgroups

subgroupG/H
centerC2C2×AGL(2,ℤ/4)
commutator, cosocle#46996⋊C3C23
FrattiniC22×C4C22×#1020
FittingC2×#8932885D3
socleC23C2×#87524
hypercenterC22AGL(2,ℤ/4)
hypocenter#73292C2×D4
LCSG ▷ #46996⋊C3 ▷ #73292
UCSC1 ◁ C2 ◁ C22

At least 9917 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has at least 9917 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 48

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

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

Polycyclic presentation:

2^4,3,2^7|a^2=j,c^2=d,d^2=j,f^2=j,g^2=j,h^2=jl,i^2=kl,c^a=cd,c^b=ck,d^a=dj,e^a=el,e^b=el,e^c=e^2jkl,e^d=ejkl,f^c=gkl,f^e=g,g^c=fjl,g^e=fgj,g^f=gj,h^a=hjk,h^c=ij,h^d=hl,h^e=il,h^f=hl,h^g=hk,i^a=il,i^c=hjl,i^d=ijkl,i^e=hij,i^f=ik,i^g=ijkl,i^h=ij,k^e=jl,l^c=jkl,l^e=jkl

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

automorphism group of#1304

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