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#72227751

order6144
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C24×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socle, hypercenterC23#87588⋊C2
commutator, cosocle, hypocenter#72209758C22
FrattiniC25C2×#1020
Fitting#72226612D3
prime radical#72227293C2
LCSG ▷ #72209758
UCSC1 ◁ C23

25355 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 25355 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 40
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 2 generators on 40 points:

(1,3,4,8)(5,6)(10,21,23,20)(11,13,14,22)(12,24)(16,18,19,17)(25,40,38,34,29,31,36,26)(27,32,28,33,35,37,39,30);(2,3,5)(6,7,8)(9,10)(11,23,22,18,12,17)(13,21,24,20,14,19)(15,16)(27,28,38)(30,31,37)(32,33,40)(35,39,36)

Polycyclic presentation:

2^2,3,2^9|b^2=jkl,d^2=k,e^2=k,f^2=jk,g^2=jk,b^a=bl,c^b=c^2,d^a=dhl,d^b=defikl,d^c=e,e^a=eil,e^b=efghjkl,e^c=dek,e^d=ek,f^a=fhl,f^b=gijkl,f^c=g,f^d=fj,f^e=fk,g^a=gil,g^b=fhjkl,g^c=fgjk,g^d=gjk,g^e=gj,g^f=gjk,h^b=i,h^c=i,h^e=hl,h^g=hl,i^b=h,i^c=hi,i^d=il,i^f=il

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72227901

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", "l");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.10*F.11*F.12);;
gap> SetPower(coll, 4, F.11);;
gap> SetPower(coll, 5, F.11);;
gap> SetPower(coll, 6, F.10*F.11);;
gap> SetPower(coll, 7, F.10*F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.12);;
gap> SetConjugate(coll, 3, 2, F.3^2);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.12);;
gap> SetConjugate(coll, 4, 2, F.4*F.5*F.6*F.9*F.11*F.12);;
gap> SetConjugate(coll, 4, 3, F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.12);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.7*F.8*F.10*F.11*F.12);;
gap> SetConjugate(coll, 5, 3, F.4*F.5*F.11);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.12);;
gap> SetConjugate(coll, 6, 2, F.7*F.9*F.10*F.11*F.12);;
gap> SetConjugate(coll, 6, 3, F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.12);;
gap> SetConjugate(coll, 7, 2, F.6*F.8*F.10*F.11*F.12);;
gap> SetConjugate(coll, 7, 3, F.6*F.7*F.10*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.10*F.11);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10*F.11);;
gap> SetConjugate(coll, 8, 2, F.9);;
gap> SetConjugate(coll, 8, 3, F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.12);;
gap> SetConjugate(coll, 8, 7, F.8*F.12);;
gap> SetConjugate(coll, 9, 2, F.8);;
gap> SetConjugate(coll, 9, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 4, F.9*F.12);;
gap> SetConjugate(coll, 9, 6, F.9*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);