Finite Group Databasebeta

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#72248196

order4608
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

derived#10596782 ⋊ C2×C4

Subgroups

subgroupG/H
center, Frattini, hypercenterC2#72215889
commutator, hypocenter#10596782C2×C4
Fitting, socleC27D32
prime radicalC2×#87588⋊C3C2
cosocleC2×#10596782C22
LCSG ▷ #10596782
UCSC1 ◁ C2

2806 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 2806 conjugacy classes of subgroups — too many to draw. Only its 27 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) = 36

A minimal faithful permutation representation — 2 generators on 16 points:

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

Polycyclic presentation:

2^3,3^2,2^6|b^2=c,d^b=d^2,e^a=e^2,f^a=g,f^e=g,g^a=f,g^e=fg,h^b=jk,h^d=hjk,h^e=i,i^a=hi,i^b=j,i^d=ij,i^e=hi,j^a=k,j^b=i,j^d=i,j^e=k,k^a=j,k^b=hi,k^d=hi,k^e=jk

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72253623, C4×#72215889
maximal normal subgroup ofC4×#72215889
maximal quotient of#72248221

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