Finite Group Databasebeta

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#72342061

order3072
automorphism?
homologyAbelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC2×C4C2×#1020⋊C2
commutator, hypocenter#997C2×C42
FrattiniC23×C4C22×S4
FittingC2×#21483D3
socleC22C4×#1020⋊C2
prime radicalC2×#3025⋊C3C22
cosocleC2×#5950C23
hypercenterC2×C8#1020⋊C2
LCSG ▷ #997
UCSC1 ◁ C2×C4 ◁ C2×C8

2709 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 2709 conjugacy classes of subgroups — too many to draw. Only its 119 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

136 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, 2, F.4);;
gap> SetPower(coll, 3, F.5);;
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.11);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10*F.11);;
gap> SetConjugate(coll, 7, 2, F.8*F.11);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9*F.11);;
gap> SetConjugate(coll, 8, 2, F.7*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, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 6, F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.11);;
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);