#72259129

order	13872
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₃×C₁₇² ⋊ C₂×C₈
Sylow	C₁₇² ⋊ D₃×C₈
Sylow	C₃ ⋊ #72213374

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃×C₁₇²	C₂×C₈
prime radical	D₅₁⋊D₁₇	C₄
cosocle	C₃×#11923883	C₂²
LCS	G ▷ C₃×C₁₇²
UCS	C₁

152 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 102

Polycyclic presentation:

2^4,3,17^2|a^2=c,b^2=c,c^2=d,e^b=e^2,f^a=f^9,f^b=f^8,f^c=f^13,f^d=f^16,g^a=f^10g^8,g^b=g^8,g^c=g^13,g^d=g^16

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#72213374
maximal normal subgroup of	D₃×#72213374

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, 2, 2, 2, 3, 17, 17]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^9);;
gap> SetConjugate(coll, 6, 2, F.6^8);;
gap> SetConjugate(coll, 6, 3, F.6^13);;
gap> SetConjugate(coll, 6, 4, F.6^16);;
gap> SetConjugate(coll, 7, 1, F.6^10*F.7^8);;
gap> SetConjugate(coll, 7, 2, F.7^8);;
gap> SetConjugate(coll, 7, 3, F.7^13);;
gap> SetConjugate(coll, 7, 4, F.7^16);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);