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#72229124

order5832
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C33

Aliases

No aliases for this group.

Splittings

SylowH(3,𝔽3)2 ⋊ D4

Subgroups

subgroupG/H
center, hypercenterC2(C3⋊D3)⋊C32
commutator, cosocleC2×C34C22×C32
Frattini, socleC2×C32C32×D32
FittingC2×H(3,𝔽3)2C22
hypocenterC34C32×D4
LCSG ▷ C2×C34 ▷ C34
UCSC1 ◁ C2

1564 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 1564 conjugacy classes of subgroups — too many to draw. Only its 106 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 — 2 generators on 22 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

173 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72259680
maximal normal subgroup of#72259680

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