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#72243762

order4374
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C34

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#14555400 ⋊ C2
derivedC3×C92 ⋊ C2×C32

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle, hypocenterC3×C92C2×C32
Frattini, socleC32C32×C3⋊(C3⋊D3)
Fitting#14555400C2
LCSG ▷ C3×C92
UCSC1

2092 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 2092 conjugacy classes of subgroups — too many to draw. Only its 191 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) = 243

A minimal faithful permutation representation — 4 generators on 54 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

147 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72243764

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