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#72277571

order4374
automorphism?
homologyAbelianization (H1) = C2 · Schur multiplier (H2) = C1

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#14550591 ⋊ C2

Subgroups

subgroupG/H
center, hypercenterC3#13091120
commutator, Fitting, cosocle, hypocenter#14550591C2
FrattiniC34C3⋊(C3⋊D3)
socleC32#92338
LCSG ▷ #14550591
UCSC1 ◁ C3

1741 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 1741 conjugacy classes of subgroups — too many to draw. Only its 71 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) = 54

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2,3^7|c^3=h^2,b^a=b^2,c^a=c^2h,c^b=ce,d^a=d^2,d^b=dg,d^c=de^2f,e^b=eh,e^c=eh^2,f^a=fh^2,g^c=gh^2,h^a=h^2

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72277578

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