#72259183

order	5408
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Splittings

derived, Sylow

C₁₃² ⋊ C₄×C₈

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₁₃⋊#261
commutator, hypocenter	C₁₃²	C₄×C₈
Fitting	C₄×C₁₃²	C₂×C₄
socle	C₂×C₁₃²	C₄²
prime radical	C₂×C₁₃⋊D₁₃	C₂×C₄
cosocle	C₄×C₁₃⋊D₁₃	C₂²
LCS	G ▷ C₁₃²
UCS	C₁ ◁ C₄

178 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 104

Polycyclic presentation:

2^5,13^2|a^2=d,b^2=c,d^2=e,f^a=f^12,f^b=f^8,f^c=f^12,g^a=f^5g,g^b=g^8,g^c=g^12

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72259204, #72259257, #72259273, …
maximal normal subgroup of	#72259204, #72259257, #72259273, …

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