#72246243

order	4032
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

Fitting, derived	C₇×H(5,𝔽₂) ⋊ C₂×C₃²
Hall	D₇×H(5,𝔽₂) ⋊ C₃²
Sylow	C₇ ⋊ C₂×Ω⁺(4,𝔽₃)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	A₄×D₇⋊A₄
commutator, Fitting, cosocle, hypocenter	C₇×H(5,𝔽₂)	C₂×C₃²
socle	C₂×C₇	C₂×A₄²
LCS	G ▷ C₇×H(5,𝔽₂)
UCS	C₁ ◁ C₂

316 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 168

Polycyclic presentation:

2,3^2,2^4,7,2|d^2=i,e^2=i,d^b=de,d^c=de,e^b=d,e^c=d,e^d=ei,f^b=fg,f^c=efi,f^e=fi,g^b=f,g^c=degi,g^d=gi,h^a=h^6,h^b=h^2

Conjugacy classes & character table

69 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×Ω⁺(4,𝔽₃)
maximal normal subgroup of	AGL(1,𝔽₇)×Ω⁺(4,𝔽₃)

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