#72242052

order	5616
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

Fitting, socle	C₂²×C₃×C₁₃ ⋊ C₃×Dic₃
derived	C₃×A₄×C₁₃ ⋊ C₃×C₄
Sylow	C₁₃ ⋊ C₃×C₃⋊SL(2,ℤ/4)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, hypocenter	C₃×A₄×C₁₃	C₃×C₄
Fitting, socle	C₂²×C₃×C₁₃	C₃×Dic₃
prime radical	C₃×A₄×D₁₃⋊C₃	C₂
cosocle	C₃×A₄×D₁₃	C₂×C₃
LCS	G ▷ C₃×A₄×C₁₃
UCS	C₁

272 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 20
transitive degree	μ_t(G) = 156

A minimal faithful permutation representation — 3 generators on 20 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	S₄×C₃⋊AGL(1,𝔽₁₃), D₃×A₄⋊AGL(1,𝔽₁₃), C₃⋊S₄×AGL(1,𝔽₁₃), …
maximal normal subgroup of	S₄×C₃⋊AGL(1,𝔽₁₃), D₃×A₄⋊AGL(1,𝔽₁₃), C₃⋊S₄×AGL(1,𝔽₁₃)
maximal quotient of	#72242208

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