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#72226221

order4608
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

FittingC22×H(5,𝔽2) ⋊ D32

Subgroups

subgroupG/H
center, Frattini, socleC22C2×S42
commutator, cosocleC2×Ω+(4,𝔽3)C23
FittingC22×H(5,𝔽2)D32
hypercenterC23S42
hypocenterΩ+(4,𝔽3)C2×D4
LCSG ▷ C2×Ω+(4,𝔽3) ▷ Ω+(4,𝔽3)
UCSC1 ◁ C22 ◁ C23

3254 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 3254 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 96

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^4,3^2,2^5|a^2=d,g^2=k,h^2=k,b^a=bd,e^a=e^2,e^c=e^2,f^a=e^2f,f^c=f^2,g^a=gk,g^c=gk,g^e=gh,g^f=h,h^a=ghk,h^c=ghk,h^e=g,h^f=gh,h^g=hk,i^a=gi,i^c=ghj,i^e=gik,i^f=j,i^g=ik,i^h=ik,j^a=jk,j^c=hi,j^e=hjk,j^f=ij,j^h=jk

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

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", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 3, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.5^2*F.6);;
gap> SetConjugate(coll, 6, 3, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 7, 5, F.7*F.8);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 1, F.7*F.8*F.11);;
gap> SetConjugate(coll, 8, 3, F.7*F.8*F.11);;
gap> SetConjugate(coll, 8, 5, F.7);;
gap> SetConjugate(coll, 8, 6, F.7*F.8);;
gap> SetConjugate(coll, 8, 7, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.7*F.9);;
gap> SetConjugate(coll, 9, 3, F.7*F.8*F.10);;
gap> SetConjugate(coll, 9, 5, F.7*F.9*F.11);;
gap> SetConjugate(coll, 9, 6, F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.11);;
gap> SetConjugate(coll, 9, 8, F.9*F.11);;
gap> SetConjugate(coll, 10, 1, F.10*F.11);;
gap> SetConjugate(coll, 10, 3, F.8*F.9);;
gap> SetConjugate(coll, 10, 5, F.8*F.10*F.11);;
gap> SetConjugate(coll, 10, 6, F.9*F.10);;
gap> SetConjugate(coll, 10, 8, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);