#72253116

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅²×C₄⋊C₈ ⋊ C₄
Sylow	C₅² ⋊ C₈×C₄⋊C₄

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#1043
commutator	C₂×C₅²	C₂×C₄×C₈
Fitting	C₅²×C₄⋊C₈	C₄
socle	C₂²×C₅²	C₂×C₄²
prime radical	C₂²×C₅⋊D₅	C₂²×C₄
cosocle	C₂×C₄×C₅⋊D₅	C₂³
hypercenter	C₄⋊C₈	#1043
hypocenter	C₅²	C₈×C₄⋊C₄
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

1184 conjugacy classes of subgroups; only the 220 normal subgroups are shown.

normal subgroups only

This group has 1184 conjugacy classes of subgroups — too many to draw. Only its 220 normal subgroups are shown.

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

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 800

Polycyclic presentation:

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72253123, C₈×D₄×#1043, #72253177
maximal normal subgroup of	#72253123, C₈×D₄×#1043, #72253177
maximal quotient of	#72253123

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