#72247206

order	6048
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁴×C₃

Aliases

No aliases for this group.

Splittings

Hall	C₃²×ΓL(1,𝔽₈) ⋊ C₂×C₂²⋊C₄
Sylow	C₇ ⋊ C₃×C₂²⋊C₄×C₃⋊D₃

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂×C₃⋊D₃×AGL(1,𝔽₇)
commutator	C₂×C₃²×C₇	C₂²×C₃×C₄
Fitting	C₂³×C₃²×C₇	C₂²×C₃
socle, cosocle	C₂²×C₃²×C₇	C₂³×C₃
prime radical	C₂³×C₃⋊(C₃⋊AGL(1,𝔽₇))	C₂
hypercenter	C₂³	C₃⋊D₃×AGL(1,𝔽₇)
hypocenter	C₃²×C₇	C₂×C₃×C₂²⋊C₄
LCS	G ▷ C₂×C₃²×C₇ ▷ C₃²×C₇
UCS	C₁ ◁ C₂² ◁ C₂³

1856 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 504

Polycyclic presentation:

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

Conjugacy classes & character table

222 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²⋊C₄×C₃⋊D₃×AGL(1,𝔽₇), #72269527
maximal normal subgroup of	C₂²⋊C₄×C₃⋊D₃×AGL(1,𝔽₇), #72269527

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