Finite Group Databasebeta

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#72242415

order3456
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C22×C3

Aliases

No aliases for this group.

Splittings

derivedC3×Ω+(4,𝔽3) ⋊ C22

Subgroups

subgroupG/H
center, Frattini, hypercenterC2S4×C3⋊S4
commutator, cosocle, hypocenterC3×Ω+(4,𝔽3)C22
FittingC3×H(5,𝔽2)D32
socleC2×C3S42
LCSG ▷ C3×Ω+(4,𝔽3)
UCSC1 ◁ C2

833 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 833 conjugacy classes of subgroups — too many to draw. Only its 39 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)(2,34,25,31)(4,24,6,21)(7,26,23,18)(8,33,11,32)(9,10,35,12)(13,27,19,30)(15,46,28,41)(17,20,44,22)(36,37,38,48)(39,42)(40,43)(50,51);(1,2,7,8)(3,14,18,29)(4,19,20,10)(5,12,23,13)(6,25,22,11)(9,28,27,15)(16,38,26,36)(17,45,21,47)(24,48,44,37)(30,39,35,42)(31,46,32,41)(33,43,34,40)(49,50,51);(1,3,15)(2,14,9)(4,16,39)(5,17,40)(6,24,46)(7,18,28)(8,29,27)(10,36,30)(11,37,31)(12,45,33)(13,47,34)(19,38,35)(20,26,42)(21,43,23)(22,44,41)(25,48,32)

Polycyclic presentation:

2^2,3^3,2^5|a^2=j,f^2=j,g^2=j,b^a=bj,c^a=c^2,c^b=c^2,d^a=c^2d,d^b=d^2,e^b=e^2,f^a=fj,f^b=fj,f^c=gj,f^d=fgj,g^a=fgj,g^b=fgj,g^c=fg,g^d=fj,g^f=gj,h^a=fhj,h^b=fh,h^c=fh,h^d=i,h^f=hj,i^a=ij,i^b=ghij,i^c=fgij,i^d=hi,i^g=ij

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD3×#11923454, #72242441, #72252919
maximal normal subgroup ofD3×#11923454, #72242441, #72252919

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