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#72245256

order6912
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C1

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC4C4⋊AGL(2,𝔽3)
commutatorC4×PSU(3,𝔽2)⋊C3C2×C4
FrattiniC2×C4C2×AGL(2,𝔽3)
FittingC2×C32×C8GL(2,𝔽3)
socleC2×C32C4⋊#859
prime radicalC22×PSU(3,𝔽2)⋊C3Q8
cosocleC2×C4×PSU(3,𝔽2)⋊C3C22
hypercenterC2×C8AGL(2,𝔽3)
hypocenterPSU(3,𝔽2)⋊C3#108
LCSG ▷ C4×PSU(3,𝔽2)⋊C3 ▷ C2×PSU(3,𝔽2)⋊C3 ▷ PSU(3,𝔽2)⋊C3
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8

552 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 25
transitive degreeμt(G) = 144

A minimal faithful permutation representation — 2 generators on 25 points:

(1,6,8,4,3,2)(5,7)(10,12,14,21,15,22,13,20)(11,16,17,19,18,24,23,25);(2,3)(4,9)(5,8)(10,11,13,17,15,18,14,23)(12,19,20,24,22,25,21,16)

Polycyclic presentation:

2^5,3,2^3,3^2|a^2=c,b^2=d,c^2=e,d^2=e,g^2=i,h^2=i,b^a=bd,d^a=de,f^a=f^2,g^a=gi,g^f=h,h^a=gh,h^f=ghi,h^g=hi,j^a=k,j^f=k,j^g=k,j^h=jk^2,j^i=j^2,k^a=j,k^f=j^2k^2,k^g=j^2,k^h=j^2k^2,k^i=k^2

Conjugacy classes & character table

104 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, 2, 2, 2, 3, 3]);;
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> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.9);;
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*F.9);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 1, 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, 1, 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, 1, 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);