#72226412

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ #1904

Subgroups

	subgroup	G/H
center	C₂	#13101558
commutator	C₂×C₄×C₅²	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₅²
Fitting	C₅²×#117	C₂²
socle	C₂×C₅²	#386
prime radical	C₂×#11716943	C₂
cosocle	C₂²×C₄×C₅²	C₂³
hypercenter	#117	D₅²
hypocenter	C₅²	#1904
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #117

1030 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 3 generators on 26 points:

(2,7)(4,12)(6,14)(8,15)(9,16)(10,13)(17,18)(19,20)(22,23)(24,25);(1,3)(2,6,8,9)(4,10)(5,11)(7,14,15,16)(12,13)(18,19)(20,21)(23,24)(25,26);(1,2,4,7,5,8,12,15)(3,9,10,16,11,6,13,14)(23,24)(25,26)

Polycyclic presentation:

2^7,5^2|a^2=e,b^2=d,e^2=g,b^a=bf,c^a=ceg,d^a=dg,e^c=eg,f^b=fg,h^a=h^4,h^b=h^4,h^c=h^4,i^a=h^3i,i^b=i^4,i^c=i^4

Conjugacy classes & character table

197 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226413
maximal normal subgroup of	#72226413

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