Finite Group Databasebeta

← Prev (#72341853) Next (#72341855) →

#72341854

order1536
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting#62654 ⋊ D3

Subgroups

subgroupG/H
center, socleC22#87524
commutator#5950C2×C4
FrattiniC2×C4C2×#1020
Fitting#62654D3
prime radicalC2×#87521C2
cosocleC2×#5950C22
hypercenterC4⋊C4#1020
hypocenter#997C22⋊C4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

2079 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 2079 conjugacy classes of subgroups — too many to draw. Only its 44 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 — 2 generators on 36 points:

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

Polycyclic presentation:

2^4,3,2^5|a^2=c,d^2=j,f^2=j,g^2=j,b^a=bd,d^a=dj,d^b=dj,e^b=e^2,f^b=fgij,f^e=g,g^b=gh,g^e=fgj,g^f=gj,h^e=i,h^f=hj,i^b=hi,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#72341862, #72341898, #72341928
maximal normal subgroup of#72341862, #72341898, #72341928

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