#72290098

order	16384
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂¹³ · Schur multiplier (H₂) = C₂⁷⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₄	C₂¹²
commutator, Frattini, cosocle	C₂	C₂¹³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ G
LECS	G ▷ C₂ ▷ C₁

37559364027177 conjugacy classes of subgroups; only the 37558989808527 normal subgroups are shown.

normal subgroups only

This group has 37559364027177 conjugacy classes of subgroups — too many to draw. Only its 37558989808527 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) = 256

Polycyclic presentation:

2^14|m^2=n,b^a=bn,d^c=dn,f^e=fn,h^g=hn,j^i=jn,l^k=ln

Conjugacy classes & character table

8194 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	2₊¹⁺¹⁴, 2₋¹⁺¹⁴
maximal normal subgroup of	2₊¹⁺¹⁴, 2₋¹⁺¹⁴

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", "m", "n");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 13, F.14);;
gap> SetConjugate(coll, 2, 1, F.2*F.14);;
gap> SetConjugate(coll, 4, 3, F.4*F.14);;
gap> SetConjugate(coll, 6, 5, F.6*F.14);;
gap> SetConjugate(coll, 8, 7, F.8*F.14);;
gap> SetConjugate(coll, 10, 9, F.10*F.14);;
gap> SetConjugate(coll, 12, 11, F.12*F.14);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);