#72241908

order	8112
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₁₃² ⋊ C₂²×C₃
Hall	#72214115 ⋊ C₃
Hall	#92789 ⋊ C₂×D₄
Sylow	C₁₃² ⋊ C₂×C₃×D₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×#11759818⋊C₂
commutator, socle, cosocle	C₂×C₁₃²	C₂³×C₃
Fitting	C₂²×C₁₃²	C₂²×C₃
hypercenter	C₂²	#11759818⋊C₂
hypocenter	C₁₃²	C₂×C₃×D₄
LCS	G ▷ C₂×C₁₃² ▷ C₁₃²
UCS	C₁ ◁ C₂ ◁ C₂²

248 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 53 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) = 156

Polycyclic presentation:

2^3,3,2,13^2|a^2=e,b^a=be,f^a=f^12,f^c=f^12,f^d=f^9,g^a=f^5g,g^c=g^12,g^d=f^11g^3

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72241924, #72242022
maximal normal subgroup of	#72241924, #72242022

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, 3, 2, 13, 13]);;
gap> SetPower(coll, 1, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^12);;
gap> SetConjugate(coll, 6, 3, F.6^12);;
gap> SetConjugate(coll, 6, 4, F.6^9);;
gap> SetConjugate(coll, 7, 1, F.6^5*F.7);;
gap> SetConjugate(coll, 7, 3, F.7^12);;
gap> SetConjugate(coll, 7, 4, F.6^11*F.7^3);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);