#72245890

order	4032
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃²×C₇×D₄ ⋊ C₂²
Hall	C₃²×C₇ ⋊ C₂×AGL(1,ℤ/8)
Sylow	C₃² ⋊ C₂×#6495
Sylow	C₇ ⋊ #10595100

Subgroups

	subgroup	G/H
center	C₂	C₂×#11759644
commutator, cosocle	C₃²×C₄×C₇	C₂⁴
Frattini	C₄	C₂²×(C₃⋊D₃)⋊D₇
Fitting	C₂×C₃²×C₇×D₄	C₂²
socle	C₂×C₃²×C₇	C₂²×D₄
hypercenter	C₂×D₄	(C₃⋊D₃)⋊D₇
hypocenter	C₃²×C₇	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₃²×C₄×C₇ ▷ C₂×C₃²×C₇ ▷ C₃²×C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₄

1428 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 168

Polycyclic presentation:

2^6,3^2,7|a^2=e,e^2=f,b^a=bef,c^a=cf,e^b=ef,g^a=g^2,g^d=g^2,h^a=g^2h,h^d=h^2,i^a=i^6

Conjugacy classes & character table

234 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×#10595100, #72246957, #72246994, …
maximal normal subgroup of	D₇×#10595100, #72246957, #72246994, …

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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 3, 3, 7]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 4, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.7^2*F.8);;
gap> SetConjugate(coll, 8, 4, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.9^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);