Finite Group Databasebeta

← Prev (#72269288) Next (#72269290) →

#72269289

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

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#14552182 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC3(C3⋊D3)⋊C32
commutator, cosocleC35C22×C32
Frattini, socleC33C32×D32
Fitting#14552182C22
prime radicalC3×#11757702C3
hypocenterC34C22×3−1+2
LCSG ▷ C35 ▷ C34
UCSC1 ◁ C3

1303 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 1303 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 108

A minimal faithful permutation representation — 2 generators on 27 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72269296
maximal normal subgroup of#72269296

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