Finite Group Databasebeta

← Prev (#72253120) Next (#72253122) →

#72253121

order4608
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C22×C3

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC4C4⋊(A4⋊S4)
commutatorC4×A42C2×C4
FrattiniC2×C4C2×A4⋊S4
FittingC25×C8C3⋊D3
socleC25C4⋊(C3⋊Dic3)
prime radicalC22×A42Q8
cosocleC2×C4×A42C22
hypercenterC2×C8A4⋊S4
hypocenterA42#108
LCSG ▷ C4×A42 ▷ C2×A42 ▷ A42
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8

1766 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 1766 conjugacy classes of subgroups — too many to draw. Only its 102 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)(9,11,13,20,14,21,12,19)(10,15,16,18,17,23,22,24);(2,4)(6,7)(9,10,12,16,14,17,13,22)(11,18,19,23,21,24,20,15);(2,3,4)(6,7,8)

Polycyclic presentation:

2^5,3^2,2^4|a^2=c,b^2=d,c^2=e,d^2=e,b^a=bd,d^a=de,f^a=f^2,g^a=g^2,h^a=i,h^f=i,h^g=i,i^a=h,i^f=hi,i^g=hi,j^a=hjk,j^f=k,j^g=hij,k^a=hik,k^f=jk,k^g=hk

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

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