#72216883

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#8932885 ⋊ C₃
derived	C₂²×H(5,𝔽₂) ⋊ C₂²×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#11710729
commutator, cosocle, hypocenter	C₂²×H(5,𝔽₂)	C₂²×C₃
Frattini, socle	C₂³	C₂²×C₂²⋊A₄
Fitting	#8932885	C₃
LCS	G ▷ C₂²×H(5,𝔽₂)
UCS	C₁ ◁ C₂

3665 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 3665 conjugacy classes of subgroups — too many to draw. Only its 53 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:

2^2,3,2^7|d^a=dh,d^b=dh,d^c=de,e^a=ehi,e^b=ehi,e^c=d,f^a=fh,f^c=g,f^e=fj,g^a=gi,g^c=fg,g^d=gj,g^e=gj,h^c=i,i^c=hi

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217073, #72217282, #72218466, …
maximal normal subgroup of	#72217073, #72217282, #72218466, …
maximal quotient of	#72216887, C₂×#72216883
automorphism group of	#2032

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", "j");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 3, F.4);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.7);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.6*F.7);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.9);;
gap> SetConjugate(coll, 9, 3, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);