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#72240989

order3072
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C27

Aliases

No aliases for this group.

Splittings

Fitting#6629045 ⋊ D3

Subgroups

subgroupG/H
center, socleC2C2×#11713271
commutator, cosocle#5950C24
FrattiniC22×C4C23×S4
Fitting#6629045D3
hypercenterM16#1020⋊C2
hypocenter#997C22×D4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

6187 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 6187 conjugacy classes of subgroups — too many to draw. Only its 125 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) = 32
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^5,3,2^5|a^2=e,d^2=ek,e^2=k,g^2=k,h^2=k,b^a=be,c^a=ck,d^a=dk,d^b=dek,d^c=dk,e^b=ek,f^d=f^2,g^b=gijk,g^d=ghjk,g^f=h,h^b=hik,h^d=hi,h^f=ghk,h^g=hk,i^f=j,i^g=ik,j^d=ij,j^f=ij,j^g=jk,j^h=jk

Conjugacy classes & character table

79 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72245616, C2×#72240989
maximal normal subgroup of#72245616, C2×#72240989
maximal quotient of#72342916, C2×#72240989

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