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#72242447

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

Aliases

No aliases for this group.

Splittings

derivedC3×Ω+(4,𝔽3) ⋊ C23

Subgroups

subgroupG/H
center, Frattini, hypercenterC2D3×S42
commutator, cosocle, hypocenterC3×Ω+(4,𝔽3)C23
FittingC3×H(5,𝔽2)C2×D32
socleC2×C3C2×S42
LCSG ▷ C3×Ω+(4,𝔽3)
UCSC1 ◁ C2

3133 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 3133 conjugacy classes of subgroups — too many to draw. Only its 63 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

87 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, 2, 1, F.2*F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.11);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 4, 3, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.4^2*F.5);;
gap> SetConjugate(coll, 5, 3, F.5^2);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7*F.11);;
gap> SetConjugate(coll, 7, 3, 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, 3, 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, 3, 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, 3, 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);