#72259137

order	27744
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₃×C₁₇² ⋊ C₂×C₁₆
Sylow	C₁₇² ⋊ C₂×C₃⋊C₁₆
Sylow	C₃ ⋊ #72213403

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃×C₁₇²	C₂×C₁₆
prime radical	C₃×D₁₇²	C₈
cosocle	C₃×#72213191	C₂²
LCS	G ▷ C₃×C₁₇²
UCS	C₁

192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 102

Polycyclic presentation:

2^5,3,17^2|b^2=c,c^2=d,d^2=e,f^b=f^2,g^a=g^16,g^b=g^12,g^c=g^8,g^d=g^13,g^e=g^16,h^a=g^3h,h^b=h^12,h^c=h^8,h^d=h^13,h^e=h^16

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#72213403
maximal normal subgroup of	D₃×#72213403

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