Finite Group Databasebeta

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#72342132

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

Aliases

No aliases for this group.

Splittings

Fitting#392195 ⋊ D3

Subgroups

subgroupG/H
center, socleC22#11713271
commutator#5950C22×C4
FrattiniC23×C4C22×S4
Fitting#392195D3
prime radicalC2×#11715739C2
cosocleC2×#5950C23
hypercenterC2×C8#1020⋊C2
hypocenter#997C4×D4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

2856 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 2856 conjugacy classes of subgroups — too many to draw. Only its 65 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)(5,35,19,7,23,22)(6,32,25,13,30,15)(8,33,11,9,26,17)(10,21)(12,29,31)(16,18)(20,36,28)(27,34);(1,2,4,3)(5,6,8,13)(7,12,9,20)(10,24,18,14)(11,25,19,15)(16,34,21,27)(17,31,22,28)(23,32,26,30)(29,35,36,33);(6,12)(7,9)(10,23)(11,19)(13,20)(14,29)(15,31)(16,33)(18,26)(21,35)(24,36)(25,28)(27,30)(32,34)

Polycyclic presentation:

2^5,3,2^5|b^2=d,c^2=dk,e^2=k,g^2=k,h^2=k,b^a=be,c^a=ck,c^b=cek,e^a=ek,e^b=ek,e^c=ek,f^a=f^2,g^a=gh,g^b=gik,g^c=gik,g^f=h,h^a=hk,h^b=hijk,h^c=hijk,h^f=ghk,h^g=hk,i^a=j,i^f=ij,i^h=ik,j^a=i,j^f=i,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, 2, F.4);;
gap> SetPower(coll, 3, F.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.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.11);;
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);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.11);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10*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, 1, F.10);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> SetConjugate(coll, 9, 8, F.9*F.11);;
gap> SetConjugate(coll, 10, 1, 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);