#72259238

order	5408
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₄×C₁₃² ⋊ C₂×C₄
Sylow	C₁₃² ⋊ C₂×C₄⋊C₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₁₃⋊#261
commutator, socle	C₂×C₁₃²	C₂²×C₄
Fitting	C₄×C₁₃²	C₂×C₄
prime radical	C₂×D₁₃²	C₂²
cosocle	C₂×C₁₃⋊D₁₃	C₂³
hypercenter	C₄	D₁₃⋊#261
hypocenter	C₁₃²	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₁₃² ▷ C₁₃²
UCS	C₁ ◁ C₂ ◁ C₄

262 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 57 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:

2^5,13^2|a^2=d,b^2=d,c^2=d,b^a=be,c^b=ce,f^a=f^5,f^b=f^8,f^c=f^8,f^d=f^12,g^a=f^6g^8,g^b=g^8,g^c=g^8,g^d=g^12

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×D₁₃⋊#261, #72259321, #72259327, …
maximal normal subgroup of	D₄×D₁₃⋊#261, #72259321, #72259327, …

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