#72218524

order	2240
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₂⁴×C₅×C₇ ⋊ C₂²
Hall	C₅×C₇ ⋊ C₂×#120
Sylow	C₇ ⋊ D₅×#120
Sylow	C₅ ⋊ D₇×#120

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅×D₇
commutator, socle, cosocle	C₂²×C₅×C₇	C₂⁴
Fitting	C₂⁴×C₅×C₇	C₂²
hypercenter	C₂⁴	D₅×D₇
hypocenter	C₅×C₇	C₂×#120
LCS	G ▷ C₂²×C₅×C₇ ▷ C₅×C₇
UCS	C₁ ◁ C₂² ◁ C₂⁴

1324 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 1324 conjugacy classes of subgroups — too many to draw. Only its 172 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|c^2=e,c^a=ce,c^b=cef,g^c=g^4,g^d=g^4,h^d=h^6

Conjugacy classes & character table

194 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72218525, D₅×D₇×#120
maximal normal subgroup of	#72218525, D₅×D₇×#120

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