#72243204

order	8400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Hall	#72243090 ⋊ C₃
Hall	C₅²×ΓL(1,𝔽₈) ⋊ C₂×D₄
Hall	C₅²×C₇ ⋊ C₂×C₃×D₄
Sylow	C₅² ⋊ C₂×(C₂²⋊D₇)⋊C₃
Hall	ΓL(1,𝔽₈) ⋊ #87886
Sylow	C₇ ⋊ C₃×#87886

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅²×AGL(1,𝔽₇)
commutator, Fitting, socle, cosocle	C₂×C₅²×C₇	C₂³×C₃
hypocenter	C₅²×C₇	C₂×C₃×D₄
LCS	G ▷ C₂×C₅²×C₇ ▷ C₅²×C₇
UCS	C₁ ◁ C₂

560 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72243208, AGL(1,𝔽₇)×#87886, #72243232
maximal normal subgroup of	#72243208, AGL(1,𝔽₇)×#87886, #72243232

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