#72259123

order	6936
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₃×C₁₇² ⋊ C₂×C₄
Sylow	C₁₇² ⋊ C₄×D₃
Sylow	C₃ ⋊ #72213364

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃×C₁₇²	C₂×C₄
prime radical	D₅₁⋊D₁₇	C₂
cosocle	C₃×C₁₇⋊D₁₇	C₂²
LCS	G ▷ C₃×C₁₇²
UCS	C₁

112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 102

Polycyclic presentation:

2^3,3,17^2|a^2=c,b^2=c,d^b=d^2,e^a=e^4,e^b=e^13,e^c=e^16,f^a=e^5f^13,f^b=f^13,f^c=f^16

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#72213364
maximal normal subgroup of	D₃×#72213364

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