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#72276406

order4400
automorphism?
homologyAbelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2

Aliases

No aliases for this group.

Splittings

FittingC4×C5×C11 ⋊ C4×C5
HallC5×C11⋊C5 ⋊ C4⋊C4
SylowC11 ⋊ C5×C4⋊AGL(1,𝔽5)

Subgroups

subgroupG/H
center, FrattiniC2C2×(C11⋊C5)⋊AGL(1,𝔽5)
commutator, socleC2×C5×C11C2×C4×C5
FittingC4×C5×C11C4×C5
prime radicalC2×D5×C11⋊C5C22
cosocleC2×D5×C11C22×C5
hypercenterC4(C11⋊C5)⋊AGL(1,𝔽5)
hypocenterC5×C11C5×C4⋊C4
LCSG ▷ C2×C5×C11 ▷ C5×C11
UCSC1 ◁ C2 ◁ C4

110 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 220

Polycyclic presentation:

2^2,5,2^2,5,11|a^2=e,b^2=d,b^a=be,f^b=f^3,f^d=f^4,g^b=g^10,g^c=g^9

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD4×(C11⋊C5)⋊AGL(1,𝔽5), AGL(1,𝔽5)×C4⋊AGL(1,𝔽11), C4⋊AGL(1,𝔽5)×AGL(1,𝔽11)
maximal normal subgroup ofD4×(C11⋊C5)⋊AGL(1,𝔽5), AGL(1,𝔽5)×C4⋊AGL(1,𝔽11), C4⋊AGL(1,𝔽5)×AGL(1,𝔽11)

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