#72218013

order	3780
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived	C₃²×C₅×C₇ ⋊ C₂²×C₃
Hall	C₃²×C₅×ΓL(1,𝔽₈) ⋊ C₂²
Hall	C₃²×ΓL(1,𝔽₈) ⋊ C₂×D₅
Hall	C₅×C₇ ⋊ C₃×D₃²
Sylow	C₇ ⋊ C₃×(C₃⋊D₃)⋊D₅
Sylow	C₅ ⋊ (C₃⋊AGL(1,𝔽₇))⋊D₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃²×C₅×C₇	C₂²×C₃
LCS	G ▷ C₃²×C₅×C₇
UCS	C₁

216 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 40 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:

2^2,3^3,5,7|d^b=d^2,e^a=e^2,f^a=f^4,f^b=f^4,g^a=g^6,g^b=g^6,g^c=g^4

Conjugacy classes & character table

93 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₅×(C₃⋊AGL(1,𝔽₇))⋊D₃, D₃²×C₅⋊AGL(1,𝔽₇), AGL(1,𝔽₇)×(C₃⋊D₃)⋊D₅
maximal normal subgroup of	D₅×(C₃⋊AGL(1,𝔽₇))⋊D₃, D₃²×C₅⋊AGL(1,𝔽₇), AGL(1,𝔽₇)×(C₃⋊D₃)⋊D₅

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