Finite Group Databasebeta

← Prev (#72259155) Next (#72259157) →

#72259156

order4608
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24×C3

Aliases

No aliases for this group.

Splittings

FittingC4×#585 ⋊ C3⋊D3

Subgroups

subgroupG/H
center, Frattini, socleC22C22×A4⋊S4
commutator, cosocleC2×Ω+(4,𝔽3)C23
FittingC4×#585C3⋊D3
hypercenterC42A4⋊S4
hypocenterΩ+(4,𝔽3)C2×D4
LCSG ▷ C2×Ω+(4,𝔽3) ▷ Ω+(4,𝔽3)
UCSC1 ◁ C22 ◁ C42

3731 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 3731 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 52
transitive degreeμt(G) = 192

A minimal faithful permutation representation — 3 generators on 52 points:

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

Polycyclic presentation:

2^4,3^2,2^5|a^2=d,c^2=k,g^2=k,h^2=k,b^a=bd,c^b=ck,e^b=e^2,f^b=f^2,g^b=hk,g^e=h,g^f=h,h^b=gk,h^e=ghk,h^f=ghk,h^g=hk,i^b=ghi,i^e=j,i^f=ghi,i^h=ik,j^b=hijk,j^e=ij,j^f=gjk,j^g=jk

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

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