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#72228955

order3888
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

FittingC22×#6896 ⋊ C22
Sylow#6896 ⋊ C2×D4

Subgroups

subgroupG/H
centerC2C2×#92264⋊C2
commutator, cosocleC2×C92C23×C3
Frattini, socleC2×C32C2×C3×D32
FittingC22×#6896C22
hypercenterC22#92264⋊C2
hypocenterC92C2×C3×D4
LCSG ▷ C2×C92 ▷ C92
UCSC1 ◁ C2 ◁ C22

693 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 693 conjugacy classes of subgroups — too many to draw. Only its 89 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) = 22
transitive degreeμt(G) = 108

A minimal faithful permutation representation — 3 generators on 22 points:

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

Polycyclic presentation:

2^3,3,2,3^4|f^3=h,g^3=i,b^a=be,f^a=f^2h^2,f^c=f^2h^2,f^d=fh^2,g^a=f^2g,g^c=g^2i^2,g^d=gh^2i,h^a=h^2,h^c=h^2,i^a=h^2i,i^c=i^2

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72228965
maximal normal subgroup of#72228965

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