#72226468

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷×C₅

Aliases

No aliases for this group.

Splittings

Fitting	C₂³×C₄×C₅² ⋊ C₄
Sylow	C₅² ⋊ C₂²⋊#95

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂²×#1044
commutator	C₂²×C₅²	C₂×C₄²
Fitting	C₂³×C₄×C₅²	C₄
socle	C₂³×C₅²	C₂²×C₄
prime radical	C₂⁴×C₅⋊D₅	C₂²
cosocle	C₂³×C₅⋊D₅	C₂³
hypercenter	C₂³×C₄	#1044
hypocenter	C₅²	C₂²⋊#95
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄

3538 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 22 points:

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

Polycyclic presentation:

2^7,5^2|a^2=d,b^2=e,b^a=bg,c^a=cf,h^a=h^2,h^d=h^4,i^a=h^3i^3,i^d=i^4

Conjugacy classes & character table

248 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72226469, #72226479

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