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#72341485

order1536
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22C2×#1020⋊C2
commutator, hypocenter#997C22×C4
FrattiniC24C22×S4
Fitting#20609D3
prime radicalC22×#997C22
cosocleC2×#997C23
hypercenterC23#1020⋊C2
LCSG ▷ #997
UCSC1 ◁ C22 ◁ C23

1528 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 1528 conjugacy classes of subgroups — too many to draw. Only its 51 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) = 64

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

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

Polycyclic presentation:

2^4,3,2^5|a^2=d,b^2=j,f^2=j,g^2=j,c^a=cj,e^b=e^2,f^a=fhj,f^b=gj,f^e=fg,g^a=ghij,g^b=fj,g^e=f,g^f=gj,h^b=hi,h^e=i,h^g=hj,i^e=hi,i^f=ij,i^g=ij

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341494, #72341602, #72341870
maximal normal subgroup of#72341494, #72341602, #72341870

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.4);;
gap> SetPower(coll, 2, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 2, F.7*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.6*F.10);;
gap> SetConjugate(coll, 7, 5, F.6);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 8, 7, 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);