#72243080

order	2800
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Hall	C₅²×C₇ ⋊ C₄⋊C₄
Sylow	C₅² ⋊ C₄⋊Dic₇
Sylow	C₇ ⋊ #87832

Subgroups

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

136 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 140

Polycyclic presentation:

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

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72243083, #72243105, D₇×#87832, …
maximal normal subgroup of	#72243083, #72243105, D₇×#87832, …

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