#72242208

order	22464
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₂²×C₃

Aliases

No aliases for this group.

Splittings

Fitting, socle	C₂⁴×C₁₃ ⋊ C₃×C₃⋊Dic₃
derived	A₄²×C₁₃ ⋊ C₃×C₄
Sylow	C₁₃ ⋊ C₃×A₄⋊SL(2,ℤ/4)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, hypocenter	A₄²×C₁₃	C₃×C₄
Fitting, socle	C₂⁴×C₁₃	C₃×C₃⋊Dic₃
prime radical	A₄²×D₁₃⋊C₃	C₂
cosocle	A₄²×D₁₃	C₂×C₃
LCS	G ▷ A₄²×C₁₃
UCS	C₁

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

normal subgroups only

This group has 932 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

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 156

A minimal faithful permutation representation — 3 generators on 21 points:

(2,7,11,8,4,13,5,6,9,3,10,12)(14,17,16,15)(18,21,19,20);(1,2,6,4,11,8,12,13,3,9,10,7,5)(19,20,21);(15,16,17)(19,20,21)

Polycyclic presentation:

2,3,2,3^2,13,2^4|a^2=c,d^a=d^2,e^a=e^2,f^a=f^8,f^b=f^3,f^c=f^12,g^d=h,g^e=h,h^a=gh,h^d=gh,h^e=gh,i^a=hij,i^d=j,i^e=ghi,j^a=gj,j^d=ij,j^e=gj

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₁₃)×A₄⋊S₄
maximal normal subgroup of	AGL(1,𝔽₁₃)×A₄⋊S₄

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", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 3, 2, 3, 3, 13, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^8);;
gap> SetConjugate(coll, 6, 2, F.6^3);;
gap> SetConjugate(coll, 6, 3, F.6^12);;
gap> SetConjugate(coll, 7, 4, F.8);;
gap> SetConjugate(coll, 7, 5, F.8);;
gap> SetConjugate(coll, 8, 1, F.7*F.8);;
gap> SetConjugate(coll, 8, 4, F.7*F.8);;
gap> SetConjugate(coll, 8, 5, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.10);;
gap> SetConjugate(coll, 9, 5, F.7*F.8*F.9);;
gap> SetConjugate(coll, 10, 1, F.7*F.10);;
gap> SetConjugate(coll, 10, 4, F.9*F.10);;
gap> SetConjugate(coll, 10, 5, F.7*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);