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#72341598

order1536
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting#32900 ⋊ D3

Subgroups

subgroupG/H
center, socleC22C2×#1020⋊C2
commutator, cosocleC2×#997C23
FrattiniC24C22×S4
Fitting#32900D3
prime radical#11715716C2
hypercenterC23#1020⋊C2
hypocenter#997C2×D4
LCSG ▷ C2×#997 ▷ #997
UCSC1 ◁ C22 ◁ C23

1645 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 1645 conjugacy classes of subgroups — too many to draw. Only its 39 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) = 20
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 20 points:

(1,2)(3,4)(5,16,9)(6,19,12)(7,17,14)(10,20,18);(1,2,3,4)(8,16)(11,19)(13,17)(15,20);(1,2,3,4)(5,6,7,10)(8,15,13,11)(9,18,14,12)(16,20,17,19)

Polycyclic presentation:

2^4,3,2^5|a^2=d,b^2=dj,f^2=j,g^2=j,c^a=cdj,c^b=cdj,e^b=e^2,f^a=fhj,f^b=fghij,f^e=g,g^a=gij,g^b=gi,g^e=fgj,g^f=gj,h^b=hi,h^e=i,h^g=hj,i^e=hi,i^f=ij

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341602, #72341813, #72341834
maximal normal subgroup of#72341602, #72341813, #72341834

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