#72243100

order	2800
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₅²×C₇ ⋊ C₂²
Hall	C₅²×C₇ ⋊ C₂×D₄
Sylow	C₅² ⋊ D₄×D₇
Sylow	C₇ ⋊ #87886

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₅×D₃₅
commutator, socle, cosocle	C₂×C₅²×C₇	C₂³
Fitting	C₂²×C₅²×C₇	C₂²
hypercenter	C₂²	D₅×D₃₅
hypocenter	C₅²×C₇	C₂×D₄
LCS	G ▷ C₂×C₅²×C₇ ▷ C₅²×C₇
UCS	C₁ ◁ C₂ ◁ C₂²

280 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 140

Polycyclic presentation:

2^4,5^2,7|a^2=d,b^a=bd,e^a=e^4,e^c=e^4,f^a=e^2f,f^c=f^4,g^c=g^6

Conjugacy classes & character table

241 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×#87886, #72243211
maximal normal subgroup of	D₇×#87886, #72243211

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