#72246123

order	9408
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Hall	#72215135 ⋊ C₃
Hall	C₇×ΓL(1,𝔽₈) ⋊ B(4,𝔽₂)
Sylow	C₇² ⋊ C₃×B(4,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂	#72245925
commutator, cosocle	C₂³×C₇²	C₂³×C₃
Frattini	C₂³	C₂×D₇×AGL(1,𝔽₇)
Fitting	C₂×C₇²×D₄	C₂²×C₃
socle	C₂×C₇²	C₃×#120
hypercenter	C₂×D₄	D₇×AGL(1,𝔽₇)
hypocenter	C₇²	C₃×B(4,𝔽₂)
LCS	G ▷ C₂³×C₇² ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂×D₄

857 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 857 conjugacy classes of subgroups — too many to draw. Only its 107 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:

2^3,3,2^3,7^2|a^2=g,c^a=cf,c^b=ce,e^a=eg,f^b=fg,h^a=h^6,h^c=h^6,h^d=h^4,i^a=h^6i,i^c=i^6,i^d=h^5i

Conjugacy classes & character table

242 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72246128
maximal normal subgroup of	#72246128

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