Finite Group Databasebeta

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#72269835

order2304
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C22×C3

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC2C4⋊(A4⋊S4)
commutator, cosocleC4×A42C22
FrattiniC4C2×A4⋊S4
FittingC24×C8C3⋊D3
socleC25C4⋊(C3⋊D3)
prime radicalC2×A42D4
hypercenterC8A4⋊S4
hypocenterA42Q16
LCSG ▷ C4×A42 ▷ C2×A42 ▷ A42
UCSC1 ◁ C2 ◁ C4 ◁ C8

713 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 713 conjugacy classes of subgroups — too many to draw. Only its 47 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) = 144

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72269839
maximal normal subgroup of#72269839

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, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.8);;
gap> SetConjugate(coll, 7, 5, F.8);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 1, F.7);;
gap> SetConjugate(coll, 8, 5, F.7*F.8);;
gap> SetConjugate(coll, 8, 6, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.7*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.8*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);