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#72228680

order5832
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2×C3

Aliases

No aliases for this group.

Splittings

Fitting, SylowH(3,𝔽3)2 ⋊ D4

Subgroups

subgroupG/H
center, hypercenterC1G
commutator, cosocle#13091502C22
Frattini, socleC32#10620206
Fitting, hypocenterH(3,𝔽3)2D4
LCSG ▷ #13091502 ▷ H(3,𝔽3)2
UCSC1

976 conjugacy classes of subgroups; only the 18 normal subgroups are shown.

normal subgroups only

This group has 976 conjugacy classes of subgroups — too many to draw. Only its 18 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) = 36
transitive degreeμt(G) = 162

A minimal faithful permutation representation — 2 generators on 36 points:

(1,9,10)(2,8,7,17,11,13)(3,16,4,12,18,5)(6,15)(19,22)(20,23,36,25,32,31)(21,27,34,28,29,35)(24,26,33);(1,2)(3,8,10,15)(4,11)(5,7)(6,9,13,16)(12,17,18,14)(19,20)(21,26,28,33)(22,25)(23,27,31,34)(24,30)(29,32,35,36)

Polycyclic presentation:

2^3,3^6|a^2=c,b^a=bc,d^a=df,d^b=df,d^c=d^2,e^a=eg,e^b=eg,e^c=e^2,f^a=df^2i^2,f^b=f^2,f^c=f^2i,f^d=fi,f^e=fi,g^a=eg^2h^2,g^b=g^2,g^c=g^2h,g^d=gi,g^e=gh,h^b=h^2,i^b=i^2

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72228688
maximal normal subgroup of#72228688

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