#72218607

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₇×#385 ⋊ C₂×C₃
Hall	#11731824 ⋊ C₃
Hall	ΓL(1,𝔽₈) ⋊ C₄×#120
Sylow	C₇ ⋊ C₃×C₄×#120

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂³×AGL(1,𝔽₇)
commutator	C₂²×C₇	C₂³×C₃×C₄
Fitting	C₇×#385	C₂×C₃
socle, cosocle	C₂³×C₇	C₂⁴×C₃
prime radical	C₂×#6445⋊C₃	C₂
hypercenter	#385	AGL(1,𝔽₇)
hypocenter	C₇	C₃×C₄×#120
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ #385

2448 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^4,3,2^3,7|a^2=g,d^2=f,b^a=bh,c^a=cg,i^c=i^6,i^d=i^6,i^e=i^2

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72218608, C₄×#120×AGL(1,𝔽₇)
maximal normal subgroup of	C₄×#120×AGL(1,𝔽₇)
maximal quotient of	#72218608

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