#72242184

order	7488
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting, socle	C₂⁴×C₁₃ ⋊ C₃×Dic₃
derived	C₁₃×C₂²⋊A₄ ⋊ C₃×C₄
Sylow	C₁₃ ⋊ C₃×#5936

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, hypocenter	C₁₃×C₂²⋊A₄	C₃×C₄
Fitting, socle	C₂⁴×C₁₃	C₃×Dic₃
prime radical	C₂²⋊A₄×D₁₃⋊C₃	C₂
cosocle	D₁₃×C₂²⋊A₄	C₂×C₃
LCS	G ▷ C₁₃×C₂²⋊A₄
UCS	C₁

628 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 32 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) = 104

A minimal faithful permutation representation — 2 generators on 21 points:

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

Polycyclic presentation:

2,3,2,3,13,2^4|a^2=c,d^a=d^2,e^a=e^8,e^b=e^3,e^c=e^12,f^a=g,f^d=fg,g^a=f,g^d=f,h^a=ghi,h^d=i,i^a=fgi,i^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72242208, #1020×AGL(1,𝔽₁₃)
maximal normal subgroup of	#1020×AGL(1,𝔽₁₃)
maximal quotient of	#72259510, #72276146

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