#72218985

order	10500
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, derived, Hall	C₇×H(3,𝔽₅) ⋊ C₃×C₄
Hall	#72218984 ⋊ C₃
Hall	ΓL(1,𝔽₈)×H(3,𝔽₅) ⋊ C₄
Sylow	H(3,𝔽₅) ⋊ ΓL(1,𝔽₈)⋊C₄
Hall	ΓL(1,𝔽₈) ⋊ #92539
Sylow	C₇ ⋊ C₃×#92539

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₅	ΓL(1,𝔽₈)⋊#1044
commutator, Fitting, hypocenter	C₇×H(3,𝔽₅)	C₃×C₄
socle	C₅×C₇	C₃×#1044
prime radical	ΓL(1,𝔽₈)×#6953	C₂
cosocle	C₇×#6953	C₂×C₃
LCS	G ▷ C₇×H(3,𝔽₅)
UCS	C₁ ◁ C₅

116 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#92539
maximal normal subgroup of	AGL(1,𝔽₇)×#92539

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