Finite Group Databasebeta

← Prev (#72241502) Next (#72241504) →

#72241503

order4608
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C23×C3

Aliases

No aliases for this group.

Splittings

FittingQ8×#120 ⋊ C3⋊D3
derivedSL(2,𝔽3)×C22⋊A4 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC2#72233459
commutator, cosocle, hypocenterSL(2,𝔽3)×C22⋊A4C22
Frattini, socleC23C2×A4⋊S4
FittingQ8×#120C3⋊D3
LCSG ▷ SL(2,𝔽3)×C22⋊A4
UCSC1 ◁ C2

1864 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 1864 conjugacy classes of subgroups — too many to draw. Only its 38 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) = 48

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

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

Polycyclic presentation:

2^2,3^2,2^7|g^2=k,h^2=k,c^a=c^2,d^a=d^2,e^a=ef,e^b=ej,e^d=f,f^b=fi,f^d=ef,g^a=hk,g^c=h,h^a=gk,h^c=ghk,h^g=hk,i^d=ij,j^a=ij,j^d=i

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofGL(2,𝔽3)×#1020⋊C2
maximal normal subgroup ofGL(2,𝔽3)×#1020⋊C2

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