#72220623

order	12168
automorphism	?
homology	Abelianization (H₁) = C₂×C₃²×C₄ · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived, Sylow	C₁₃² ⋊ C₂×C₃²×C₄
Hall	C₁₃⋊C₃² ⋊ C₂×C₄
Hall	D₁₃⋊#261 ⋊ C₃²

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₁₃²	C₂×C₃²×C₄
prime radical	D₁₃⋊C₃²	C₂
cosocle	C₁₃⋊D₁₃	C₂²×C₃²
LCS	G ▷ C₁₃²
UCS	C₁

204 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

2^2,3^2,2,13^2|b^2=e,f^a=f^12,f^b=f^8,f^c=f^9,f^d=f^3,f^e=f^12,g^a=f^11g,g^b=g^8,g^c=f^8g,g^d=g^3,g^e=g^12

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72220623, AGL(1,𝔽₁₃)²
maximal normal subgroup of	C₂×#72220623, AGL(1,𝔽₁₃)²
maximal quotient of	#72241943, C₂×#72220623, #72242029, …
holomorph of	D₁₃⋊C₃

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, 3, 2, 13, 13]);;
gap> SetPower(coll, 2, F.5);;
gap> SetConjugate(coll, 6, 1, F.6^12);;
gap> SetConjugate(coll, 6, 2, F.6^8);;
gap> SetConjugate(coll, 6, 3, F.6^9);;
gap> SetConjugate(coll, 6, 4, F.6^3);;
gap> SetConjugate(coll, 6, 5, F.6^12);;
gap> SetConjugate(coll, 7, 1, F.6^11*F.7);;
gap> SetConjugate(coll, 7, 2, F.7^8);;
gap> SetConjugate(coll, 7, 3, F.6^8*F.7);;
gap> SetConjugate(coll, 7, 4, F.7^3);;
gap> SetConjugate(coll, 7, 5, F.7^12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);