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#72245359

order3456
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C1

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC2C4⋊AGL(2,𝔽3)
commutator, cosocleC4×PSU(3,𝔽2)⋊C3C22
FrattiniC4C2×AGL(2,𝔽3)
FittingC32×C8GL(2,𝔽3)
socleC2×C32C4⋊GL(2,𝔽3)
prime radicalC2×PSU(3,𝔽2)⋊C3D4
hypercenterC8AGL(2,𝔽3)
hypocenterPSU(3,𝔽2)⋊C3Q16
LCSG ▷ C4×PSU(3,𝔽2)⋊C3 ▷ C2×PSU(3,𝔽2)⋊C3 ▷ PSU(3,𝔽2)⋊C3
UCSC1 ◁ C2 ◁ C4 ◁ C8

288 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 26 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)(10,12,19,23,14,21,13,20)(11,15,22,25,17,24,16,18);(2,3)(4,9)(5,8)(10,11,14,17)(12,18,21,25)(13,22,19,16)(15,23,24,20)

Polycyclic presentation:

2^4,3,2^3,3^2|a^2=d,b^2=cd,c^2=d,f^2=h,g^2=h,b^a=bc,c^a=cd,e^a=e^2,f^a=fh,f^e=g,g^a=fg,g^e=fgh,g^f=gh,i^a=j,i^e=j,i^f=j,i^g=ij^2,i^h=i^2,j^a=i,j^e=i^2j^2,j^f=i^2,j^g=i^2j^2,j^h=j^2

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72245405
maximal normal subgroup of#72245405

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