Finite Group Databasebeta

← Prev (#72343864) Next (#72343866) →

#72343865

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

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×#11714184C2
cosocleC2×#5950C23
hypercenterC2×C8#1020⋊C2
hypocenter#997C2×C22⋊C4
LCSG ▷ #5950 ▷ #997
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

3791 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 3791 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:

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

Polycyclic presentation:

2^5,3,2^5|a^2=d,c^2=dk,d^2=k,e^2=k,b^a=be,c^a=ck,c^b=cek,d^b=dk,e^b=ek,f^c=f^2,g^b=gik,g^c=ghijk,g^f=h,h^b=hjk,h^c=hjk,h^f=gh,i^c=ij,i^f=j,i^h=ik,j^f=ij,j^g=jk

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72343876, #72343956, #72343985
maximal normal subgroup of#72343876, #72343956, #72343985

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, 3, F.4*F.11);;
gap> SetPower(coll, 4, F.11);;
gap> SetPower(coll, 5, 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, 4, 2, F.4*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*F.9*F.10*F.11);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 2, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 6, F.7*F.8);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 6, F.10);;
gap> SetConjugate(coll, 9, 8, F.9*F.11);;
gap> SetConjugate(coll, 10, 6, F.9*F.10);;
gap> SetConjugate(coll, 10, 7, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);