#72218893

order	18496
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₁₇² ⋊ C₄⋊C₁₆

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	#72213403
commutator, Fitting, socle	C₂×C₁₇²	C₂×C₁₆
prime radical	C₂×C₁₇⋊D₁₇	C₂×C₈
cosocle	C₂×#72213191	C₂²
hypocenter	C₁₇²	C₄⋊C₁₆
LCS	G ▷ C₂×C₁₇² ▷ C₁₇²
UCS	C₁ ◁ C₂

212 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^6,17^2|a^2=d,b^2=c,c^2=e,e^2=f,b^a=bd,g^a=g^16,g^b=g^12,g^c=g^8,g^e=g^13,g^f=g^16,h^a=g^3h,h^b=h^12,h^c=h^8,h^e=h^13,h^f=h^16

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72218906
maximal normal subgroup of	#72218906

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 17, 17]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 7, 1, F.7^16);;
gap> SetConjugate(coll, 7, 2, F.7^12);;
gap> SetConjugate(coll, 7, 3, F.7^8);;
gap> SetConjugate(coll, 7, 5, F.7^13);;
gap> SetConjugate(coll, 7, 6, F.7^16);;
gap> SetConjugate(coll, 8, 1, F.7^3*F.8);;
gap> SetConjugate(coll, 8, 2, F.8^12);;
gap> SetConjugate(coll, 8, 3, F.8^8);;
gap> SetConjugate(coll, 8, 5, F.8^13);;
gap> SetConjugate(coll, 8, 6, F.8^16);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);