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#72243743

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

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#14554867 ⋊ C22

Subgroups

subgroupG/H
center, socle, hypercenterC3(C3⋊D3)⋊C32
commutator, cosocle, hypocenterH(5,𝔽3)C22×C32
FrattiniC33C32×D32
Fitting#14554867C22
LCSG ▷ H(5,𝔽3)
UCSC1 ◁ C3

920 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 920 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^2,3^7|c^3=i^2,d^3=i^2,d^c=di^2,e^a=e^2,e^b=e^2,e^c=eg,e^d=eg^2,f^a=e^2f,f^b=f^2,f^c=fg^2i,f^d=fg^2h,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

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72243757
maximal normal subgroup of#72243757

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, 3, F.9^2);;
gap> SetPower(coll, 4, F.9^2);;
gap> SetConjugate(coll, 4, 3, F.4*F.9^2);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.7^2);;
gap> SetConjugate(coll, 6, 1, F.5^2*F.6);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 6, 3, F.6*F.7^2*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.7^2*F.8);;
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);