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#72341894

order3072
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Splittings

Fitting#392717 ⋊ D3

Subgroups

subgroupG/H
center, socleC22#11713271
commutator#5950C22×C4
FrattiniC23×C4C22×S4
Fitting#392717D3
prime radicalC2×#11714174C2
cosocleC2×#5950C23
hypercenterC2×C8#1020⋊C2
hypocenter#997C2×C22⋊C4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

3872 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 3872 conjugacy classes of subgroups — too many to draw. Only its 69 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:

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

Polycyclic presentation:

2^5,3,2^5|b^2=d,e^2=k,g^2=k,h^2=k,c^b=ce,e^b=ek,e^c=ek,f^a=f^2,g^a=ghjk,g^b=gijk,g^c=gijk,g^f=h,h^a=hi,h^b=hik,h^c=hik,h^f=ghk,h^g=hk,i^f=j,i^g=ik,j^a=ij,j^f=ij,j^g=jk,j^h=jk

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72341940
maximal normal subgroup of#72341940

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 5, F.11);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.10*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.11);;
gap> SetConjugate(coll, 8, 6, F.7*F.8*F.11);;
gap> SetConjugate(coll, 8, 7, F.8*F.11);;
gap> SetConjugate(coll, 9, 6, F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.11);;
gap> SetConjugate(coll, 10, 1, F.9*F.10);;
gap> SetConjugate(coll, 10, 6, F.9*F.10);;
gap> SetConjugate(coll, 10, 7, F.10*F.11);;
gap> SetConjugate(coll, 10, 8, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);