#72260975

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	C₂²×#3549 ⋊ C₃
derived	C₂²×H(3,𝔽₄) ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₂²×H(3,𝔽₄)	C₂×C₃
Frattini	C₂²	C₂×C₂²⋊(C₂²⋊A₄)
Fitting	C₂²×#3549	C₃
socle	C₂⁴	C₂×C₂²⋊A₄
LCS	G ▷ C₂²×H(3,𝔽₄)
UCS	C₁

14232 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 14232 conjugacy classes of subgroups — too many to draw. Only its 95 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) = 48

Polycyclic presentation:

2,3,2^8|e^2=j,f^2=i,b^a=bij,c^b=d,d^b=cd,e^b=fj,f^b=ef,g^a=gj,g^b=h,g^e=gj,g^f=gij,h^a=hi,h^b=gh,h^e=hij,h^f=hi,i^b=ij,j^b=i

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72261629, #72262232, #72340794, …
maximal subgroup of	#72260994, #72261016, #72261348, …
maximal normal subgroup of	#72260994, #72261016, #72261348, …

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