#72238850

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂×#120
commutator	C₄²	C₂×C₄³
Frattini, cosocle	C₂³×C₄²	C₂⁴
socle	C₂³	C₂×#1437
prime radical	C₂⁶	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂⁴×C₄² ◁ G
LECS	G ▷ C₂³×C₄² ▷ C₂² ▷ C₁

21332 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6,9,15,11,14,10,16)(7,12,17,19,8,13,18,20)(21,22)(23,27)(24,30)(25,29)(26,31)(28,32)(33,35)(34,36);(2,4)(5,7,10,18,11,8,9,17)(6,12,16,20,14,13,15,19)(21,23,24,32,26,34,25,33)(22,27,30,28,31,36,29,35);(1,2,3,4)(5,8,11,7)(6,13,14,12)(9,18,10,17)(15,20,16,19)(21,23,26,34)(22,28,31,35)(24,32,25,33)(27,30,36,29);(6,14)(7,8)(15,16)(17,18)(21,24,26,25)(22,29,31,30)(23,32,34,33)(27,35,36,28)

Polycyclic presentation:

2^11|a^2=e,b^2=f,c^2=ghjk,d^2=k,e^2=j,f^2=jk,g^2=jk,h^2=jk,i^2=k,c^a=ci,c^b=cg,d^a=djk,d^b=dj,d^c=dj,g^a=gk,g^b=gjk,g^c=gjk,h^a=hk,h^b=hjk,h^c=hjk,i^a=ik,i^b=ik,i^c=ik

Conjugacy classes & character table

320 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238851
maximal normal subgroup of	#72238851

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 3, F.7*F.8*F.10*F.11);;
gap> SetPower(coll, 4, F.11);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.10*F.11);;
gap> SetPower(coll, 7, F.10*F.11);;
gap> SetPower(coll, 8, F.10*F.11);;
gap> SetPower(coll, 9, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.10*F.11);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.10*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.10*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> SetConjugate(coll, 8, 2, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.10*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> SetConjugate(coll, 9, 2, F.9*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);