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AGL(1,𝔽97) #72210660

order9312
automorphism?
homologyAbelianization (H1) = C3×C32 · Schur multiplier (H2) = C1

Aliases

affine general linearAGL(1,𝔽97)

Splittings

Fitting, socle, derived, SylowC97 ⋊ C3×C32
Hall#15690911 ⋊ C3
HallC97⋊C3 ⋊ C32

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC97C3×C32
prime radicalD97⋊C3C16
cosocle#13095525C2×C3
LCSG ▷ C97
UCSC1

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2,3,2^4,97|a^2=c,c^2=d,d^2=e,e^2=f,g^a=g^28,g^b=g^35,g^c=g^8,g^d=g^64,g^e=g^22,g^f=g^96

Conjugacy classes & character table

97 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×AGL(1,𝔽97), #72218955, C2×AGL(1,𝔽97)
maximal normal subgroup ofC3×AGL(1,𝔽97), C2×AGL(1,𝔽97)
maximal quotient ofC3×AGL(1,𝔽97), #72218955, C2×AGL(1,𝔽97)
automorphism group ofD97
holomorph ofC97

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, 3, 2, 2, 2, 2, 97]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 7, 1, F.7^28);;
gap> SetConjugate(coll, 7, 2, F.7^35);;
gap> SetConjugate(coll, 7, 3, F.7^8);;
gap> SetConjugate(coll, 7, 4, F.7^64);;
gap> SetConjugate(coll, 7, 5, F.7^22);;
gap> SetConjugate(coll, 7, 6, F.7^96);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);