Finite Group Databasebeta

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#72243790

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

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#14554854 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, Fitting, cosocle, hypocenter#14554854C22
FrattiniC33D3×C3⋊(C3⋊D3)
socleC3#72214354
LCSG ▷ #14554854
UCSC1

2186 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 2186 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 27
transitive degreeμt(G) = 27

A minimal faithful permutation representation — 4 generators on 27 points:

(1,2)(3,19)(4,7)(5,12)(8,10)(9,17)(11,27)(13,25)(14,15)(16,23)(18,22)(20,24)(21,26);(2,14,12,6,7,15)(3,20,13,9,11,21)(10,25,23,19,22,27);(1,8,17)(2,9,10)(3,19,6)(4,18,24)(5,16,26)(7,20,22)(11,25,14)(12,21,23)(13,27,15);(2,7,12)(6,14,15)(8,18,16)(17,24,26)

Polycyclic presentation:

2^2,3^7|c^b=c^2,d^a=d^2,d^b=d^2,d^c=dg,e^a=d^2e,e^b=e^2,e^c=eh,f^b=f^2,f^d=fg^2,f^e=fgi^2,g^a=g^2i,g^b=gi^2,g^d=gi,g^e=gi^2,h^a=g^2hi^2,h^d=hi^2,i^b=i^2

Conjugacy classes & character table

91 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72243867, #72244038
maximal normal subgroup of#72243867, #72244038

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