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#72344170

order1536
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C24

Aliases

No aliases for this group.

Splittings

Fitting#7734 ⋊ D3
derived#2679⋊C3 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC2#11711670
commutator, cosocle, hypocenter#2679⋊C3C22
FrattiniC2×C42C2×S4
Fitting#7734D3
socleC23#1020⋊C2
LCSG ▷ #2679⋊C3
UCSC1 ◁ C2

642 conjugacy classes of subgroups; only the 13 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 13 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) = 24
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 2 generators on 24 points:

(1,6)(2,3,7,13)(4,16,15,17)(5,18)(8,12,9,14)(10,21,24,23,20,22,19,11);(3,10)(6,11)(12,24)(13,19)(14,20)(16,21)(17,22)(18,23)

Polycyclic presentation:

2^2,3,2^7|f^2=i,g^2=ij,c^a=c^2,c^b=c^2,d^a=e,d^b=efhij,d^c=e,e^a=d,e^b=dfgh,e^c=de,f^a=fgij,f^b=fgi,f^c=gi,f^d=fh,f^e=fi,g^a=gij,g^c=fg,g^d=ghj,g^e=ghi,h^c=hi,i^a=j,i^b=j,i^c=ij,j^a=i,j^b=i,j^c=i

Conjugacy classes & character table

31 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72344171
maximal normal subgroup of#72344171

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