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#72277543

order4374
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C38

Aliases

No aliases for this group.

Splittings

Fitting, derived, SylowC32⋊H(5,𝔽3) ⋊ C2

Subgroups

subgroupG/H
center, Frattini, socle, hypercenterC32C3⋊(C3⋊(C3⋊(C3⋊D3)))
commutator, Fitting, cosocle, hypocenterC32⋊H(5,𝔽3)C2
LCSG ▷ C32⋊H(5,𝔽3)
UCSC1 ◁ C32

12748 conjugacy classes of subgroups; only the 2700 normal subgroups are shown.

normal subgroups only

This group has 12748 conjugacy classes of subgroups — too many to draw. Only its 2700 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) = 36
transitive degreeμt(G) = 81

A minimal faithful permutation representation — 6 generators on 36 points:

(2,4)(5,9)(6,8)(11,14)(12,18)(15,24)(16,26)(17,22)(19,30)(20,29)(23,27)(25,33)(28,35)(31,36)(32,34);(12,19,20)(16,25,28)(18,29,30)(23,32,31)(26,35,33)(27,36,34);(10,12,18)(11,16,27)(13,20,29)(14,23,26)(15,25,34)(17,28,36)(19,30,21)(22,31,35)(24,32,33);(2,5,6)(4,8,9)(11,15,17)(14,22,24)(16,25,28)(23,31,32)(26,35,33)(27,34,36);(1,2,4)(3,6,8)(5,9,7);(10,11,14)(12,16,23)(13,17,22)(15,24,21)(18,27,26)(19,25,32)(20,28,31)(29,36,35)(30,34,33)

Polycyclic presentation:

2,3^7|b^a=b^2,c^a=c^2,d^a=d^2,d^b=dg,d^c=dh,e^a=e^2,f^a=f^2,f^e=fg

Conjugacy classes & character table

171 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72277678, #72277745
maximal normal subgroup of#72277678, #72277745

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