Finite Group Databasebeta

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#72226200

order3072
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C8

Aliases

No aliases for this group.

Splittings

SylowC3 ⋊ #72214006

Subgroups

subgroupG/H
centerC22#10688954
commutatorC3×C82C22×C4
FrattiniC2×C82C22×D3
FittingC3×#122211C2
socleC22×C3#8813
prime radicalC23×C3#1705
cosocleC2×C3×C82C23
hypercenter#122211D3
hypocenterC3#72214006
LCSG ▷ C3×C82 ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ #122211

846 conjugacy classes of subgroups; only the 383 normal subgroups are shown.

normal subgroups only

This group has 846 conjugacy classes of subgroups — too many to draw. Only its 383 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) = 35
transitive degreeμt(G) = 768

A minimal faithful permutation representation — 3 generators on 35 points:

(1,2,3)(20,21,22,25,23,27,28,32,24,26,29,31,30,33,35,34);(2,3)(5,10,13,15)(6,8)(7,16)(11,19,18,12)(14,17)(21,25,26,31)(22,28)(23,30)(27,34,33,32)(29,35);(2,3)(4,5,6,11,9,13,8,18)(7,15,14,19,17,10,16,12)(21,25,26,31)(22,28)(23,30)(27,34,33,32)(29,35)

Polycyclic presentation:

2^10,3|a^2=d,b^2=dg,c^2=f,d^2=i,e^2=g,f^2=h,g^2=i,h^2=j,b^a=be,c^a=cf,c^b=cf,d^b=di,d^c=dj,e^a=eg,e^b=eg,f^a=fh,f^b=fh,g^a=gi,g^b=gi,h^a=hj,h^b=hj,k^a=k^2,k^b=k^2

Conjugacy classes & character table

348 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, 2, 2, 2, 2, 2, 2, 3]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.4*F.7);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.9);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 11, 1, F.11^2);;
gap> SetConjugate(coll, 11, 2, F.11^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);