#72247173

order	3024
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂²×C₃

Aliases

No aliases for this group.

Splittings

Hall	C₃²×ΓL(1,𝔽₈) ⋊ C₂²⋊C₄
Sylow	C₇ ⋊ C₃×#3757

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂×C₃⋊(C₃⋊AGL(1,𝔽₇))
commutator	C₂×C₃²×C₇	C₂×C₃×C₄
Fitting	C₂³×C₃²×C₇	C₂×C₃
socle, cosocle	C₂²×C₃²×C₇	C₂²×C₃
prime radical	C₂³×C₃²×ΓL(1,𝔽₈)	C₂
hypercenter	C₂³	C₃⋊(C₃⋊AGL(1,𝔽₇))
hypocenter	C₃²×C₇	C₃×C₂²⋊C₄
LCS	G ▷ C₂×C₃²×C₇ ▷ C₃²×C₇
UCS	C₁ ◁ C₂² ◁ C₂³

664 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 504

Polycyclic presentation:

2^2,3,2^2,3^2,7|a^2=d,b^a=be,f^a=f^2,g^a=g^2,h^a=h^6,h^c=h^4

Conjugacy classes & character table

198 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃⋊D₃×(C₂²⋊Dic₇)⋊C₃, C₂²⋊C₄×C₃⋊(C₃⋊AGL(1,𝔽₇)), AGL(1,𝔽₇)×#3757, …
maximal normal subgroup of	C₃⋊D₃×(C₂²⋊Dic₇)⋊C₃, C₂²⋊C₄×C₃⋊(C₃⋊AGL(1,𝔽₇)), AGL(1,𝔽₇)×#3757, …

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