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#72268992

order3456
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, FrattiniC22C2×AGL(2,𝔽3)
commutatorC2×PSU(3,𝔽2)⋊C3C2×C4
FittingC2×C32×C4GL(2,𝔽3)
socleC22×C32C2×GL(2,𝔽3)
prime radical, cosocleC22×PSU(3,𝔽2)⋊C3C22
hypercenterC2×C4AGL(2,𝔽3)
hypocenterPSU(3,𝔽2)⋊C3C4⋊C4
LCSG ▷ C2×PSU(3,𝔽2)⋊C3 ▷ PSU(3,𝔽2)⋊C3
UCSC1 ◁ C22 ◁ C2×C4

458 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 17
transitive degreeμt(G) = 144

A minimal faithful permutation representation — 2 generators on 17 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC4⋊C4×AGL(2,𝔽3)
maximal normal subgroup ofC4⋊C4×AGL(2,𝔽3)
maximal quotient of#72269011

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.3);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 2, 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, 2, 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, 2, 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);