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#72243740

order8748
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C32

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#14554343 ⋊ C22

Subgroups

subgroupG/H
center, socleC3C3×#11757702
commutator, cosocle, hypocenterH(5,𝔽3)C22×C32
FrattiniC33C32×D32
Fitting#14554343C22
hypercenterC32#11757702
LCSG ▷ H(5,𝔽3)
UCSC1 ◁ C3 ◁ C32

1520 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 1520 conjugacy classes of subgroups — too many to draw. Only its 75 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) = 81
transitive degreeμt(G) = 81

A minimal faithful permutation representation — 2 generators on 81 points:

(1,13,8,16,7,36,5,9,24,15,2,28,6,30,22,3,11,25)(4,60,26,72,23,79,21,70,33,39,10,77,20,67,47,71,19,80)(12,61,57,35,27,65,42,34,53)(14,45,64,50,48,75,40,52,58,46,37,56,41,32,54,18,29,63)(17,59,62,38,43,81,49,74,73,68,51,78,44,66,55,69,31,76);(1,14,38)(2,37,74)(3,4,18,12,17,39)(5,40,68)(6,41,69)(7,48,59)(8,64,78)(9,23,45,27,51,70)(10,52,61,31,67,30)(11,29,66)(13,19,32,34,43,60)(15,20,46,35,44,72)(16,21,50,42,49,71)(22,54,81)(24,58,76)(25,33,75,53,62,79)(26,63,57,55,80,28)(36,47,56,65,73,77)

Polycyclic presentation:

2^2,3^7|d^3=i,d^c=di,e^a=e^2,e^b=e^2,e^d=eg,f^a=e^2f,f^b=f^2,f^d=fgh^2,g^a=g^2i,g^b=g^2i,g^e=gi,g^f=gi^2,h^a=g^2hi^2,h^b=h^2i^2,h^e=hi^2

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72243757, #72253861
maximal normal subgroup of#72243757, #72253861

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