Finite Group Databasebeta

← Prev (#72259801) Next (#72259803) →

#72259802

order15552
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C3

Aliases

No aliases for this group.

Splittings

derivedC32×PSU(3,𝔽2)⋊C3 ⋊ C2×C4

Subgroups

subgroupG/H
center, Frattini, hypercenterC2D3×C3⋊AGL(2,𝔽3)
commutator, hypocenterC32×PSU(3,𝔽2)⋊C3C2×C4
Fitting, socleC2×C34C2×GL(2,𝔽3)
prime radicalC2×#72213922C2
cosocleC2×C32×PSU(3,𝔽2)⋊C3C22
LCSG ▷ C32×PSU(3,𝔽2)⋊C3
UCSC1 ◁ C2

2351 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 2351 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 19
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 3 generators on 19 points:

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

Polycyclic presentation:

2^3,3^3,2^3,3^2|a^2=c,g^2=i,h^2=i,d^a=d^2,d^b=d^2,e^a=d^2e,e^b=e^2,f^b=f^2,g^b=gi,g^f=h,h^b=gh,h^f=ghi,h^g=hi,j^b=k,j^f=k,j^g=k,j^h=jk^2,j^i=j^2,k^b=j,k^f=j^2k^2,k^g=j^2,k^h=j^2k^2,k^i=k^2

Conjugacy classes & character table

126 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, 3, 3, 3, 2, 2, 2, 3, 3]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.4^2*F.5);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 2, F.7*F.8);;
gap> SetConjugate(coll, 8, 6, F.7*F.8*F.9);;
gap> SetConjugate(coll, 8, 7, F.8*F.9);;
gap> SetConjugate(coll, 10, 2, F.11);;
gap> SetConjugate(coll, 10, 6, F.11);;
gap> SetConjugate(coll, 10, 7, F.11);;
gap> SetConjugate(coll, 10, 8, F.10*F.11^2);;
gap> SetConjugate(coll, 10, 9, F.10^2);;
gap> SetConjugate(coll, 11, 2, F.10);;
gap> SetConjugate(coll, 11, 6, F.10^2*F.11^2);;
gap> SetConjugate(coll, 11, 7, F.10^2);;
gap> SetConjugate(coll, 11, 8, F.10^2*F.11^2);;
gap> SetConjugate(coll, 11, 9, F.11^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);