#72225731

order	4096
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂×#2846
commutator, Frattini, cosocle	C₂⁷	C₂⁵
LCS	G ▷ C₂⁷ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁸ ◁ G
LECS	G ▷ C₂⁷ ▷ C₂² ▷ C₁

At least 14872 conjugacy classes of subgroups; only the 2495 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

(5,6)(7,8)(17,18)(19,23)(20,22)(21,24);(1,2)(3,4)(9,10)(11,15)(12,14)(13,16);(13,15)(21,23);(2,4)(6,8)(13,15)(21,23);(9,11)(10,13)(12,16)(14,15)(17,19)(18,21)(20,24)(22,23)

Polycyclic presentation:

2^12|c^a=cj,c^b=ch,d^a=di,d^b=dg,e^a=el,e^b=ek,e^c=ef,e^d=ef,f^a=fl,f^b=fk,g^e=gk,h^e=hk,i^e=il,j^e=jl

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226388, #72256496, #72329442, …
maximal normal subgroup of	#72226388, #72329442, #72334245, …
automorphism group of	C₂×C₄×D₄, C₂×#121, C₂×C₂²⋊Q₈

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", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.12);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.6);;
gap> SetConjugate(coll, 5, 4, F.5*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.12);;
gap> SetConjugate(coll, 6, 2, F.6*F.11);;
gap> SetConjugate(coll, 7, 5, F.7*F.11);;
gap> SetConjugate(coll, 8, 5, F.8*F.11);;
gap> SetConjugate(coll, 9, 5, F.9*F.12);;
gap> SetConjugate(coll, 10, 5, F.10*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);