#72243191

order	16800
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Hall	#72243083 ⋊ C₃
Hall	C₅²×ΓL(1,𝔽₈) ⋊ C₄×D₄
Hall	C₅²×C₇ ⋊ C₃×C₄×D₄
Sylow	C₅² ⋊ C₄×C₄⋊AGL(1,𝔽₇)
Hall	ΓL(1,𝔽₈) ⋊ #11717375
Sylow	C₇ ⋊ C₃×#11717375

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	AGL(1,𝔽₇)×D₅⋊AGL(1,𝔽₅)
commutator, Fitting, socle	C₂×C₅²×C₇	C₂²×C₃×C₄
prime radical	#72243182	C₂
cosocle	C₂×C₇×C₅⋊D₅	C₂³×C₃
hypocenter	C₅²×C₇	C₃×C₄×D₄
LCS	G ▷ C₂×C₅²×C₇ ▷ C₅²×C₇
UCS	C₁ ◁ C₂

968 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 968 conjugacy classes of subgroups — too many to draw. Only its 115 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) = 140

Polycyclic presentation:

2^3,3,2^2,5^2,7|a^2=f,c^2=e,b^a=bf,g^a=g^4,g^c=g^3,g^e=g^4,h^a=g^3h,h^c=h^3,h^e=h^4,i^b=i^6,i^d=i^4

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#11717375
maximal normal subgroup of	AGL(1,𝔽₇)×#11717375

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