#72259277

order	10816
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂×D₄×C₁₃² ⋊ C₂²
Sylow	C₁₃² ⋊ C₂×AGL(1,ℤ/8)

Subgroups

	subgroup	G/H
center	C₂	C₂×#72214115
commutator, cosocle	C₄×C₁₃²	C₂⁴
Frattini	C₄	C₂²×D₁₃²
Fitting	C₂×D₄×C₁₃²	C₂²
socle	C₂×C₁₃²	C₂²×D₄
hypercenter	C₂×D₄	D₁₃²
hypocenter	C₁₃²	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₁₃² ▷ C₂×C₁₃² ▷ C₁₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₄

1304 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 1304 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 104

Polycyclic presentation:

2^6,13^2|a^2=e,e^2=f,b^a=bef,c^a=cf,e^b=ef,g^a=g^12,g^d=g^12,h^a=g^5h,h^d=h^12

Conjugacy classes & character table

514 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72259289, #72259514
maximal normal subgroup of	#72259289, #72259514

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