#72215751

order	8802
automorphism	?
homology	Abelianization (H₁) = C₂×C₂₇ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Sylow	C₁₆₃ ⋊ C₂×C₂₇
Hall	#17984252 ⋊ C₂
Hall	D₁₆₃ ⋊ C₂₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₁₆₃	C₂×C₂₇
prime radical	D₁₆₃⋊C₃	C₉
cosocle	#13091814	C₂×C₃
LCS	G ▷ C₁₆₃
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2,3^3,163|b^3=c,c^3=d,e^a=e^162,e^b=e^115,e^c=e^85,e^d=e^104

Conjugacy classes & character table

57 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₁₆₃)
maximal normal subgroup of	AGL(1,𝔽₁₆₃)

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 3, 3, 3, 163]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 5, 1, F.5^162);;
gap> SetConjugate(coll, 5, 2, F.5^115);;
gap> SetConjugate(coll, 5, 3, F.5^85);;
gap> SetConjugate(coll, 5, 4, F.5^104);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);