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#72271968

order1536
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C22

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#87524
commutator, cosocle#3529⋊C3C22
FrattiniC2×C4C2×#1020
Fitting#63045D3
prime radical#63045⋊C3C2
hypercenterC2×D4#1020
hypocenter#5465D4
LCSG ▷ #3529⋊C3 ▷ #5465
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

1125 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 1125 conjugacy classes of subgroups — too many to draw. Only its 29 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) = 48
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 2 generators on 48 points:

(1,16,14,15,5,9,4,6)(2,11,12,3,8,13,7,10)(17,21,22,33)(18,36,27,38)(19,45,28,30)(20,32)(23,26,24,43)(25,37)(31,34)(35,41)(39,42)(40,48);(3,4,13)(6,7,16)(9,15,12)(10,14,11)(17,18)(19,23)(20,26,34,25,21,42)(22,27)(24,28)(29,36,46,35,30,48)(31,41,33,40,32,43)(37,45,39,44,38,47)

Polycyclic presentation:

2^3,3,2^6|b^2=ij,c^2=i,e^2=i,f^2=i,g^2=ij,h^2=ij,b^a=bc,c^a=ci,c^b=ci,d^b=d^2,e^b=ef,e^d=f,f^b=fi,f^d=efi,f^e=fi,g^b=gh,g^d=h,g^f=gj,h^b=hij,h^d=ghij,h^e=hj,h^g=hij

Conjugacy classes & character table

43 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72271983
maximal normal subgroup of#72271983

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