#72246558

order	12096
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵×C₃×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃²×C₇×D₄ ⋊ C₂²×C₃
Hall	C₃²×ΓL(1,𝔽₈) ⋊ C₂×#127
Sylow	C₇ ⋊ C₃×C₃⋊D₃×#127

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂²×C₃⋊D₃×AGL(1,𝔽₇)
commutator, socle, cosocle	C₂²×C₃²×C₇	C₂⁴×C₃
Fitting	C₂×C₃²×C₇×D₄	C₂²×C₃
hypercenter	C₂×D₄	C₃⋊D₃×AGL(1,𝔽₇)
hypocenter	C₃²×C₇	C₂×C₃×#127
LCS	G ▷ C₂²×C₃²×C₇ ▷ C₃²×C₇
UCS	C₁ ◁ C₂² ◁ C₂×D₄

6840 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 1008

Polycyclic presentation:

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

Conjugacy classes & character table

306 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃⋊D₃×#127×AGL(1,𝔽₇)
maximal normal subgroup of	C₃⋊D₃×#127×AGL(1,𝔽₇)

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", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 2, 2, 3, 3, 7]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> SetConjugate(coll, 8, 3, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> SetConjugate(coll, 9, 3, F.9^2);;
gap> SetConjugate(coll, 10, 3, F.10^6);;
gap> SetConjugate(coll, 10, 5, F.10^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);