#72218906

order	36992
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₁₇² ⋊ C₂×C₁₆
Sylow	C₁₇² ⋊ D₄×C₁₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×#72213403
commutator, socle	C₂×C₁₇²	C₂²×C₁₆
Fitting	C₂²×C₁₇²	C₂×C₁₆
prime radical	#72218884	C₈
cosocle	C₂×#72213191	C₂³
hypercenter	C₂²	#72213403
hypocenter	C₁₇²	D₄×C₁₆
LCS	G ▷ C₂×C₁₇² ▷ C₁₇²
UCS	C₁ ◁ C₂ ◁ C₂²

740 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

2^7,17^2|a^2=e,c^2=d,d^2=f,f^2=g,b^a=be,h^a=h^16,h^c=h^12,h^d=h^8,h^f=h^13,h^g=h^16,i^a=hi,i^c=i^12,i^d=i^8,i^f=i^13,i^g=i^16

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

Relationships

holomorph of

Dic₁₇, C₂×D₁₇

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 17, 17]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 8, 1, F.8^16);;
gap> SetConjugate(coll, 8, 3, F.8^12);;
gap> SetConjugate(coll, 8, 4, F.8^8);;
gap> SetConjugate(coll, 8, 6, F.8^13);;
gap> SetConjugate(coll, 8, 7, F.8^16);;
gap> SetConjugate(coll, 9, 1, F.8*F.9);;
gap> SetConjugate(coll, 9, 3, F.9^12);;
gap> SetConjugate(coll, 9, 4, F.9^8);;
gap> SetConjugate(coll, 9, 6, F.9^13);;
gap> SetConjugate(coll, 9, 7, F.9^16);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);