#72222280

order	2720
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Hall	C₅×C₁₇ ⋊ C₂×C₁₆
Sylow	C₁₇ ⋊ D₅⋊C₁₆
Sylow	C₅ ⋊ C₂×AGL(1,𝔽₁₇)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅×C₁₇	C₂×C₁₆
prime radical	D₅×D₁₇	C₈
cosocle	#10621779	C₂²
LCS	G ▷ C₅×C₁₇
UCS	C₁

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^5,5,17|a^2=c,b^2=d,c^2=d,d^2=e,f^a=f^2,f^c=f^4,g^a=g^12,g^b=g^8,g^c=g^8,g^d=g^13,g^e=g^16

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×AGL(1,𝔽₁₇), #72244745, C₂×#72222280, …
maximal normal subgroup of	AGL(1,𝔽₅)×AGL(1,𝔽₁₇), C₂×#72222280
maximal quotient of	#72244745, C₂×#72222280, #72244811, …

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