#72243649

order	23520
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Hall	#72243483 ⋊ C₃
Hall	C₅×#3861 ⋊ C₂×C₂²⋊C₄
Hall	C₅×C₇² ⋊ C₂×C₃×C₂²⋊C₄
Hall	#3861 ⋊ C₂×C₂²⋊AGL(1,𝔽₅)
Sylow	C₇² ⋊ C₂×C₃×C₂²⋊AGL(1,𝔽₅)
Sylow	C₅ ⋊ #72214840

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	AGL(1,𝔽₅)×#64423⋊C₂
commutator, Fitting, socle	C₂×C₅×C₇²	C₂²×C₃×C₄
prime radical	C₂×D₅×#64423⋊C₂	C₂
cosocle	C₂×C₇²×D₅	C₂³×C₃
hypocenter	C₅×C₇²	C₂×C₃×C₂²⋊C₄
LCS	G ▷ C₂×C₅×C₇² ▷ C₅×C₇²
UCS	C₁ ◁ C₂

1380 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 140

Polycyclic presentation:

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

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×#72210376
maximal normal subgroup of	AGL(1,𝔽₅)×#72210376

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