#72276641

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Hall	#11729215 ⋊ C₃
Hall	ΓL(1,𝔽₈) ⋊ #2177
Sylow	C₇ ⋊ C₃×#2177

Subgroups

	subgroup	G/H
center	C₂²	D₈×AGL(1,𝔽₇)
commutator	C₇×C₈	C₂²×C₃×C₄
Frattini	C₂×C₈	C₂²×AGL(1,𝔽₇)
Fitting	C₂×C₇×D₁₆	C₂×C₃
socle	C₂²×C₇	C₂×C₃×D₈
prime radical	C₂×ΓL(1,𝔽₈)×D₁₆	C₂
cosocle	C₂×C₇×C₈	C₂³×C₃
hypercenter	C₂×D₁₆	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#2177
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₁₆

344 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 105 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) = 448

Polycyclic presentation:

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

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

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