Finite Group Databasebeta

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#72225203

order2048
nilpotency class5
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#123285
commutatorC8×C16C22×C4
Frattini, cosocleC2×C8×C16C23
prime radicalC2×#12515C22
LCSG ▷ C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C8×C16 ◁ G
LECSG ▷ C2×C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

854 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 854 conjugacy classes of subgroups — too many to draw. Only its 140 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) = 128

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

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

Polycyclic presentation:

2^11|a^2=f,c^2=fi,d^2=gi,e^2=hj,g^2=i,h^2=jk,j^2=k,c^a=ce,c^b=cd,d^a=dh,d^b=dg,d^c=dg,e^a=eh,e^b=eh,e^c=ehk,g^a=gj,g^b=gi,g^c=gi,h^a=hj,h^b=hj,h^c=hj,i^a=ik,j^a=jk,j^b=jk,j^c=jk

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72225219, #72226950
maximal normal subgroup of#72225219, #72226950
maximal quotient of#72227154

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