#72259265

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

Aliases

No aliases for this group.

Splittings

Fitting	D₄×C₁₃² ⋊ C₂²
Sylow	C₁₃² ⋊ C₂×D₈

Subgroups

	subgroup	G/H
center	C₂	#72214115
commutator, cosocle	C₄×C₁₃²	C₂³
Frattini	C₄	C₂×D₁₃²
Fitting	D₄×C₁₃²	C₂²
socle	C₂×C₁₃²	C₂×D₄
hypercenter	D₄	D₁₃²
hypocenter	C₁₃²	C₂×D₈
LCS	G ▷ C₄×C₁₃² ▷ C₂×C₁₃² ▷ C₁₃²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₄

314 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 40 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,d^2=e,b^a=bd,d^b=de,f^a=f^12,f^c=f^12,g^a=f^2g,g^c=g^12

Conjugacy classes & character table

278 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72259273, #72259277, #72259283, …
maximal normal subgroup of	#72259273, #72259277, #72259283, …

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