#72259447

order	8112
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting	D₄×C₁₃² ⋊ C₂×C₃
Hall	C₁₃×#6104 ⋊ C₃
Hall	#92788 ⋊ D₈
Sylow	C₁₃² ⋊ C₃×D₈

Subgroups

	subgroup	G/H
center	C₂	#72214085
commutator, cosocle	C₄×C₁₃²	C₂²×C₃
Frattini	C₄	C₂×#11759817
Fitting	D₄×C₁₃²	C₂×C₃
socle	C₂×C₁₃²	C₃×D₄
hypercenter	D₄	#11759817
hypocenter	C₁₃²	C₃×D₈
LCS	G ▷ C₄×C₁₃² ▷ C₂×C₁₃² ▷ C₁₃²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₄

184 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 29 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:

2^2,3,2^2,13^2|a^2=d,d^2=e,b^a=bd,d^b=de,f^a=f^12,f^c=f^3,g^a=f^8g,g^c=g^3

Conjugacy classes & character table

179 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72259450, #72259483, #72259490
maximal normal subgroup of	#72259450, #72259483, #72259490

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