#72242153

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ #1313

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×#1043
commutator	C₄×C₅²	C₄×C₈
Frattini	C₂×C₈	C₂×#1043
Fitting	C₂×C₅²×C₁₆	C₄
socle, prime radical	C₂²×C₅²	#97
cosocle	C₈×#1039	C₂²
hypercenter	C₂×C₁₆	#1043
hypocenter	C₅²	#1313
LCS	G ▷ C₄×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₁₆

320 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 116 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) = 3200

Polycyclic presentation:

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

Conjugacy classes & character table

236 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72242158
maximal normal subgroup of	#72242158

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