#72218209

order	2240
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Splittings

Hall	C₅×C₇ ⋊ C₄×C₂²⋊C₄
Sylow	C₇ ⋊ C₂²⋊C₄×AGL(1,𝔽₅)
Sylow	C₅ ⋊ C₂²⋊C₄×Dic₇

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂×D₇×AGL(1,𝔽₅)
commutator	C₂×C₅×C₇	C₂×C₄²
Fitting	C₂³×C₅×C₇	C₂×C₄
socle	C₂²×C₅×C₇	C₂²×C₄
prime radical	C₂³×C₇×D₅	C₂²
cosocle	C₂²×C₇×D₅	C₂³
hypercenter	C₂³	D₇×AGL(1,𝔽₅)
hypocenter	C₅×C₇	C₄×C₂²⋊C₄
LCS	G ▷ C₂×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂² ◁ C₂³

516 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 130 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|b^2=ef,c^2=d,b^a=be,g^c=g^3,g^d=g^4,h^b=h^6,h^c=h^6

Conjugacy classes & character table

134 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72218210, D₇×C₂²⋊C₄×AGL(1,𝔽₅), #72218600, …
maximal normal subgroup of	#72218210, D₇×C₂²⋊C₄×AGL(1,𝔽₅), #72218600, …

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