#72226518

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ #3021

Subgroups

	subgroup	G/H
center	C₂	C₂×#11717095
commutator, cosocle	C₂×C₄×C₅²	C₂⁴
Frattini	C₂×C₄	C₂²×D₅²
Fitting	C₂×C₅²×#41	C₂²
socle	C₂×C₅²	C₂×#120
hypercenter	C₂×#41	D₅²
hypocenter	C₅²	#3021
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#41

1806 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 26 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

221 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226522
maximal normal subgroup of	#72226522

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