#72223329

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ #1914

Subgroups

	subgroup	G/H
center	C₂	#13101549
commutator	C₂³×C₅²	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₅²
Fitting	C₅²×#117	C₂²
socle	C₂×C₅²	#386
prime radical	#13107056	C₂
cosocle	C₂²×C₄×C₅²	C₂³
hypercenter	#117	D₅²
hypocenter	C₅²	#1914
LCS	G ▷ C₂³×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #117

1014 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 1014 conjugacy classes of subgroups — too many to draw. Only its 124 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

2^7,5^2|a^2=e,e^2=g,b^a=bf,c^b=cd,d^a=dg,e^b=eg,f^a=fg,f^c=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	#72223330
maximal normal subgroup of	#72223330

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, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, 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, 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);