#72258992

order	3072
automorphism	?
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

H(5,𝔽₂)² ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, socle, hypercenter	C₂²	C₂²⋊(C₂²⋊(C₂²⋊A₄))
commutator, Fitting, cosocle, hypocenter	H(5,𝔽₂)²	C₃
LCS	G ▷ H(5,𝔽₂)²
UCS	C₁ ◁ C₂²

169518 conjugacy classes of subgroups; only the 546 normal subgroups are shown.

normal subgroups only

This group has 169518 conjugacy classes of subgroups — too many to draw. Only its 546 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) = 64

Polycyclic presentation:

3,2^10|b^a=c,c^a=bc,d^a=e,d^b=dj,d^c=dj,e^a=de,e^c=ej,f^a=g,f^d=fj,g^a=fg,g^d=gj,g^e=gj,h^a=i,h^b=hj,h^c=hj,h^f=hk,h^g=hk,i^a=hi,i^c=ij,i^g=ik

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72258996, #72259022, #72259091
maximal subgroup of	#72258996, #72259006, #72259022
maximal normal subgroup of	#72258996, #72259006, #72259022

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", "k");;
gap> coll := SingleCollector(F, [3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetConjugate(coll, 2, 1, F.3);;
gap> SetConjugate(coll, 3, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, 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, 1, F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.11);;
gap> SetConjugate(coll, 8, 7, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.8*F.9);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);