#72259496

order	4992
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₁₃×H(5,𝔽₂) ⋊ Dic₃
derived	C₁₃×#997 ⋊ C₄
Sylow	C₁₃ ⋊ #85565

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	#72242150
commutator, hypocenter	C₁₃×#997	C₄
Fitting	C₁₃×H(5,𝔽₂)	Dic₃
socle	C₂×C₁₃	#5936
prime radical, cosocle	D₁₃×#997	C₂
LCS	G ▷ C₁₃×#997
UCS	C₁ ◁ C₂

398 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 23 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:

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

Polycyclic presentation:

2^2,3,13,2^5|a^2=b,e^2=i,f^2=i,c^a=c^2,d^a=d^8,d^b=d^12,e^a=ef,e^c=f,f^a=fi,f^c=efi,f^e=fi,g^a=gh,g^c=h,g^f=gi,h^c=gh,h^e=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72259510, #261×#5935
maximal normal subgroup of	#72259510, #261×#5935

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