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#72277600

order4374
automorphism?
homologyAbelianization (H1) = C2×C33 · Schur multiplier (H2) = C34

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#14555252 ⋊ C2

Subgroups

subgroupG/H
centerC3C3×#92391
commutator, cosocle, hypocenterC3×H(3,𝔽3)C2×C33
Frattini, socleC32C33×C3⋊D3
Fitting#14555252C2
hypercenterC32#92391
LCSG ▷ C3×H(3,𝔽3)
UCSC1 ◁ C3 ◁ C32

3760 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 3760 conjugacy classes of subgroups — too many to draw. Only its 203 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) = 54
transitive degreeμt(G) = 54

A minimal faithful permutation representation — 3 generators on 54 points:

(1,2)(3,7)(4,42,20,17,15,45)(5,47,25,21,22,49)(6,11)(8,41,33,18,31,43)(9,52,36,24,34,50)(10,26)(12,27)(13,28)(14,32,40,35,38,54)(16,30)(19,51,44,46,23,29)(37,53)(39,48);(1,5,24)(2,9,21)(3,20,19)(4,44,13)(6,25,52)(7,51,17)(8,35,48)(10,36,47)(11,34,49)(12,14,43)(15,23,16)(18,37,40)(22,50,26)(27,31,54)(28,46,42)(29,45,30)(32,53,33)(38,41,39);(1,14,44)(2,9,49)(3,25,18)(4,24,12)(5,43,13)(6,40,19)(7,29,17)(8,35,27)(10,36,21)(11,34,47)(15,50,39)(16,22,41)(20,52,37)(23,26,38)(28,51,42)(30,46,45)(31,54,53)(32,48,33)

Polycyclic presentation:

2,3^7|c^b=ch,e^a=e^2,e^c=eg,e^d=eg,f^a=f^2,f^d=fg,f^e=fh,g^a=g^2

Conjugacy classes & character table

183 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72277661

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