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#72228256

order2304
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C24×C3

Aliases

No aliases for this group.

Splittings

FittingC24×D4 ⋊ C3⋊D3

Subgroups

subgroupG/H
center, FrattiniC2C22×A4⋊S4
commutator, cosocleC2×A42C23
FittingC24×D4C3⋊D3
socleC25C22×C3⋊D3
hypercenterD4A4⋊S4
hypocenterA42#41
LCSG ▷ C2×A42 ▷ A42
UCSC1 ◁ C2 ◁ D4

2340 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 2340 conjugacy classes of subgroups — too many to draw. Only its 77 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) = 16
transitive degreeμt(G) = 72

A minimal faithful permutation representation — 3 generators on 16 points:

(1,4)(2,3)(5,8,7)(9,11,12,16)(10,14,13,15);(3,4)(6,7)(10,13)(14,15);(2,4)(7,8)(9,10,12,13)(11,15,16,14)

Polycyclic presentation:

2^4,3^2,2^4|a^2=d,c^2=d,b^a=bd,c^a=cd,e^a=e^2,e^b=e^2,f^a=f^2,f^b=f^2,g^e=h,g^f=h,h^a=gh,h^b=gh,h^e=gh,h^f=gh,i^a=hij,i^b=hij,i^e=j,i^f=ghi,j^a=gj,j^b=gj,j^e=ij,j^f=gj

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72258808, #41×A4⋊S4
maximal normal subgroup of#72258808, #41×A4⋊S4

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