#72259285

order	10816
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄×C₁₃

Aliases

No aliases for this group.

Splittings

Fitting	C₂×D₄×C₁₃² ⋊ C₄
Sylow	C₁₃² ⋊ #418

Subgroups

	subgroup	G/H
center	C₂	C₂×#72214162
commutator	C₄×C₁₃²	C₂²×C₄
Frattini	C₄	C₂²×#10621729
Fitting	C₂×D₄×C₁₃²	C₄
socle	C₂×C₁₃²	C₂×C₂²⋊C₄
prime radical	C₂×D₄×C₁₃⋊D₁₃	C₂
cosocle	C₄×C₁₃⋊D₁₃	C₂³
hypercenter	C₂×D₄	#10621729
hypocenter	C₁₃²	#418
LCS	G ▷ C₄×C₁₃² ▷ C₂×C₁₃² ▷ C₁₃²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₄

1088 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 68 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=d,d^2=f,e^2=f,b^a=be,c^a=cf,d^b=df,e^b=ef,g^a=g^5,g^d=g^12,h^a=g^6h^8,h^d=h^12

Conjugacy classes & character table

442 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72259289, #72259572
maximal normal subgroup of	#72259289, #72259572

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