Finite Group Databasebeta

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#72341911

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

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, socleC22#11713200
commutator#5950C22×C4
FrattiniC23×C4C22×S4
FittingC2×#32894D3
prime radicalC2×#32894⋊C3C2
cosocleC2×#5950C23
hypercenterC2×D4#1020⋊C2
hypocenter#997C4×D4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

5259 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 5259 conjugacy classes of subgroups — too many to draw. Only its 81 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:

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

Polycyclic presentation:

2^5,3,2^5|c^2=d,e^2=k,g^2=k,h^2=k,b^a=be,c^b=ck,e^a=ek,e^b=ek,f^c=f^2,g^b=gik,g^c=gh,g^f=h,h^b=hijk,h^c=hk,h^f=ghk,h^g=hk,i^c=j,i^f=ij,i^h=ik,j^c=i,j^f=i,j^g=jk,j^h=jk

Conjugacy classes & character table

82 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, 3, F.4);;
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.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.11);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 6, 3, F.6^2);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*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, 3, F.10);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> SetConjugate(coll, 9, 8, F.9*F.11);;
gap> SetConjugate(coll, 10, 3, F.9);;
gap> SetConjugate(coll, 10, 6, F.9);;
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);