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#72341906

order1536
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC4#73081
commutator#5950C2×C4
FrattiniC23×C4C2×S4
Fitting#60398D3
socleC2#11711332
prime radical#60398⋊C3C2
cosocleC2×#5950C22
hypercenterC2×C4#1020⋊C2
hypocenter#997C22⋊C4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C4 ◁ C2×C4

961 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 961 conjugacy classes of subgroups — too many to draw. Only its 33 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 — 2 generators on 32 points:

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

Polycyclic presentation:

2^4,3,2^5|a^2=c,b^2=cd,c^2=j,d^2=j,b^a=bd,d^a=dj,e^a=e^2,e^b=e^2,f^a=g,f^b=ghj,f^e=g,g^a=f,g^b=fhij,g^e=fg,h^a=hi,h^b=hi,h^e=i,h^f=hj,i^e=hi,i^f=ij,i^g=ij

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341913
maximal normal subgroup of#72341913

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