#72223100

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅²×#106 ⋊ C₄
Sylow	C₅² ⋊ #1941

Subgroups

	subgroup	G/H
center	C₂²	D₄×#1043
commutator	C₂×C₄×C₅²	C₂²×C₄
Frattini	C₂×C₄	C₂²×#1043
Fitting	C₅²×#106	C₄
socle	C₂²×C₅²	C₄×D₄
prime radical	C₂²×C₅⋊D₅	C₂×D₄
cosocle	C₂×C₄×C₅⋊D₅	C₂³
hypercenter	#106	#1043
hypocenter	C₅²	#1941
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

992 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 992 conjugacy classes of subgroups — too many to draw. Only its 166 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,b^2=d,c^2=ef,d^2=g,f^2=g,b^a=bf,c^a=cef,d^a=dg,f^a=fg,h^b=h^3,h^d=h^4,i^b=i^3,i^d=i^4

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#498×#1043, #72223344, #72223350
maximal normal subgroup of	#498×#1043, #72223344, #72223350

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