#72218327

order	2240
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₅×C₇×D₄ ⋊ C₂×C₄
Hall	C₅×C₇ ⋊ #417
Sylow	C₇ ⋊ #65793
Sylow	C₅ ⋊ #89543

Subgroups

	subgroup	G/H
center	C₂	D₇×C₂²⋊AGL(1,𝔽₅)
commutator	C₄×C₅×C₇	C₂²×C₄
Frattini	C₄	C₂×D₇×AGL(1,𝔽₅)
Fitting	C₅×C₇×D₄	C₂×C₄
socle	C₂×C₅×C₇	C₂×C₂²⋊C₄
prime radical	D₅×#1159	C₂
cosocle	C₄×C₇×D₅	C₂³
hypercenter	D₄	D₇×AGL(1,𝔽₅)
hypocenter	C₅×C₇	#417
LCS	G ▷ C₄×C₅×C₇ ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₄

324 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 68 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:

2^6,5,7|a^2=d,b^2=def,d^2=f,e^2=f,b^a=be,c^a=cf,d^b=df,e^b=ef,g^a=g^3,g^b=g^3,g^d=g^4,h^c=h^6

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72218329, AGL(1,𝔽₅)×#6474, #72218536, …
maximal normal subgroup of	#72218329, AGL(1,𝔽₅)×#6474, #72218536, …

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