Finite Group Databasebeta

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#72341840

order1536
automorphism?
homologyAbelianization (H1) = C42 · Schur multiplier (H2) = C2×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, hypercenterC2×C4#1020⋊C2
commutator, hypocenter#997C42
FrattiniC23×C4C2×S4
Fitting#20608D3
socleC22#73080
prime radical, cosocleC2×#5950C22
LCSG ▷ #997
UCSC1 ◁ C2×C4

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

normal subgroups only

This group has 865 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) = 36
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 2 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341847, #72341862, #72341881, …
maximal normal subgroup of#72341847, #72341862, #72341881, …

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