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#72243822

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

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#14554861 ⋊ C22

Subgroups

subgroupG/H
center, socle, hypercenterC3(C3⋊D3)⋊C32
commutator, cosocle, hypocenterH(5,𝔽3)C22×C32
FrattiniC33C32×D32
Fitting#14554861C22
LCSG ▷ H(5,𝔽3)
UCSC1 ◁ C3

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

normal subgroups only

This group has 984 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72243840, #72244030
maximal normal subgroup of#72243840, #72244030

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, 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);;
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.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.7^2*F.9);;
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);;
gap> SetConjugate(coll, 8, 5, F.8*F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);