Finite Group Databasebeta

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#72252547

order3072
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting#392135 ⋊ D3

Subgroups

subgroupG/H
center, socleC2#72220366
commutatorC2×#5950C2×C4
FrattiniC23×D4C2×S4
Fitting#392135D3
prime radical#72227639C2
cosocleC2×#87547C22
hypercenterC2×D4#1020⋊C2
hypocenter#997#99
LCSG ▷ C2×#5950 ▷ C2×#997 ▷ #997
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4

4570 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 4570 conjugacy classes of subgroups — too many to draw. Only its 33 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) = 32
transitive degreeμt(G) = 32

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

2^5,3,2^5|a^2=d,b^2=ek,c^2=k,g^2=k,h^2=k,b^a=bc,c^a=ce,d^b=dek,d^c=dk,e^a=ek,f^b=f^2,g^a=gjk,g^b=gh,g^f=h,h^a=hik,h^b=hk,h^f=ghk,h^g=hk,i^f=ij,i^g=ik,j^b=ij,j^f=i,j^h=jk

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72252563, #72267056, #72267075
maximal normal subgroup of#72252563, #72267056, #72267075

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