#72226487

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ #2290

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₅⋊AGL(1,𝔽₅)
commutator, socle	C₂²×C₅²	C₂³×C₄
Frattini	C₂²	C₂²×D₅⋊AGL(1,𝔽₅)
Fitting	C₂²×C₄×C₅²	C₂×C₄
prime radical	C₂×#87886	C₂²
cosocle	C₂²×C₅⋊D₅	C₂⁴
hypercenter	C₂²×C₄	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	#2290
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

1890 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 4 generators on 22 points:

(1,2,5,3,4)(6,7,10,9,8)(15,17,18,21)(16,20,19,22);(2,3,4,5)(7,9,8,10)(16,19)(20,22);(12,14);(7,8)(9,10)(11,12)(13,14)(15,16,18,19)(17,20,21,22)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226493
maximal normal subgroup of	#72226493

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