#72244257

order	4032
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×C₇×D₄ ⋊ C₂²×C₃
Hall	C₃×ΓL(1,𝔽₈) ⋊ C₂×#127
Sylow	C₇ ⋊ C₃×D₃×#127

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃×AGL(1,𝔽₇)
commutator, socle, cosocle	C₂²×C₃×C₇	C₂⁴×C₃
Fitting	C₂×C₃×C₇×D₄	C₂²×C₃
hypercenter	C₂×D₄	D₃×AGL(1,𝔽₇)
hypocenter	C₃×C₇	C₂×C₃×#127
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₂×D₄

2100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 336

Polycyclic presentation:

2^4,3,2^2,3,7|d^2=g,c^a=cf,d^a=dg,d^c=dg,h^b=h^2,i^a=i^6,i^b=i^6,i^e=i^4

Conjugacy classes & character table

150 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#127×AGL(1,𝔽₇), #72246558, #72254457, …
maximal normal subgroup of	D₃×#127×AGL(1,𝔽₇), #72254457, #72254465, …
maximal quotient of	#72246558

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