#72259284

order	5408
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	D₄×C₁₃² ⋊ C₂²
Sylow	C₁₃² ⋊ H(5,𝔽₂)

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂²×D₁₃²
commutator, socle, cosocle	C₂×C₁₃²	C₂⁴
Fitting	D₄×C₁₃²	C₂²
hypercenter	D₄	D₁₃²
hypocenter	C₁₃²	H(5,𝔽₂)
LCS	G ▷ C₂×C₁₃² ▷ C₁₃²
UCS	C₁ ◁ C₂ ◁ D₄

494 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 108 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=e,c^a=ce,d^a=de,d^b=de,f^b=f^12,f^d=f^12,g^b=f^5g,g^d=g^12

Conjugacy classes & character table

317 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72259301, #72259641
maximal normal subgroup of	#72259301, #72259641

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