Finite Group Databasebeta

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#72252930

order6912
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

FittingC2×C3×H(5,𝔽2) ⋊ D32
derivedC3×Ω+(4,𝔽3) ⋊ C23

Subgroups

subgroupG/H
center, FrattiniC2C2×#13181790
commutator, cosocle, hypocenterC3×Ω+(4,𝔽3)C23
FittingC2×C3×H(5,𝔽2)D32
socleC2×C3C2×S42
hypercenterC22#13181790
LCSG ▷ C3×Ω+(4,𝔽3)
UCSC1 ◁ C2 ◁ C22

3147 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 3147 conjugacy classes of subgroups — too many to draw. Only its 67 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) = 51
transitive degreeμt(G) = 144

A minimal faithful permutation representation — 3 generators on 51 points:

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

Polycyclic presentation:

2^3,3^3,2^5|g^2=k,h^2=k,c^a=ck,d^a=d^2,d^b=d^2,e^a=d^2e,e^b=e^2,f^a=f^2,g^a=gk,g^b=gk,g^d=hk,g^e=ghk,h^a=ghk,h^b=ghk,h^d=gh,h^e=gk,h^g=hk,i^a=gik,i^b=gi,i^d=gi,i^e=j,i^g=ik,j^a=jk,j^b=hijk,j^d=ghjk,j^e=ij,j^h=jk

Conjugacy classes & character table

96 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, 3, 3, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.11);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.4^2*F.5);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.11);;
gap> SetConjugate(coll, 7, 4, F.8*F.11);;
gap> SetConjugate(coll, 7, 5, F.7*F.8*F.11);;
gap> SetConjugate(coll, 8, 1, F.7*F.8*F.11);;
gap> SetConjugate(coll, 8, 2, F.7*F.8*F.11);;
gap> SetConjugate(coll, 8, 4, F.7*F.8);;
gap> SetConjugate(coll, 8, 5, F.7*F.11);;
gap> SetConjugate(coll, 8, 7, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.7*F.9*F.11);;
gap> SetConjugate(coll, 9, 2, F.7*F.9);;
gap> SetConjugate(coll, 9, 4, F.7*F.9);;
gap> SetConjugate(coll, 9, 5, F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.11);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 2, F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 10, 4, F.7*F.8*F.10*F.11);;
gap> SetConjugate(coll, 10, 5, F.9*F.10);;
gap> SetConjugate(coll, 10, 8, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);