Finite Group Databasebeta

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#72342144

order3072
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting#354788 ⋊ D3

Subgroups

subgroupG/H
center, socleC22#11713272
commutator#5950C22×C4
FrattiniC23×C4C22×S4
Fitting#354788D3
prime radicalC2×#11715739C2
cosocleC2×#5950C23
hypercenterC2×C8#1020⋊C2
hypocenter#997C4×D4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

2816 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 2816 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

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, 1, F.4);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.4*F.5*F.11);;
gap> SetPower(coll, 4, F.11);;
gap> SetPower(coll, 5, F.11);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.11);;
gap> SetConjugate(coll, 6, 3, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.10*F.11);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
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, 3, 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);