#72276146

order	14976
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₁₃×H(5,𝔽₂) ⋊ C₃×Dic₃
derived	C₁₃×#997 ⋊ C₃×C₄
Sylow	C₁₃ ⋊ C₃×#85564

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	#72242184
commutator, hypocenter	C₁₃×#997	C₃×C₄
Fitting	C₁₃×H(5,𝔽₂)	C₃×Dic₃
socle	C₂×C₁₃	C₃×#5936
prime radical	D₁₃⋊C₃×#997	C₂
cosocle	D₁₃×#997	C₂×C₃
LCS	G ▷ C₁₃×#997
UCS	C₁ ◁ C₂

976 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 976 conjugacy classes of subgroups — too many to draw. Only its 39 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:

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

Polycyclic presentation:

2,3,2,3,13,2^5|a^2=c,d^a=d^2,e^a=e^8,e^b=e^3,e^c=e^12,f^a=g,f^d=g,g^a=f,g^d=fg,h^a=i,h^d=i,h^g=hj,i^a=h,i^d=hi,i^f=ij

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₁₃)×#5934
maximal normal subgroup of	AGL(1,𝔽₁₃)×#5934

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, 13, 2, 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.7);;
gap> SetConjugate(coll, 7, 1, F.6);;
gap> SetConjugate(coll, 7, 4, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.9);;
gap> SetConjugate(coll, 8, 4, F.9);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.8);;
gap> SetConjugate(coll, 9, 4, F.8*F.9);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);