#72226836

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₈²
commutator	C₈²	C₂³×C₄
Frattini, cosocle	C₂×C₈²	C₂⁴
prime radical	C₂²×D₁₆	D₈
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

1506 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 1024

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

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

Polycyclic presentation:

2^11|b^2=e,c^2=fhj,d^2=gi,f^2=hj,g^2=ik,h^2=j,i^2=k,b^a=bk,c^a=cg,c^b=cf,d^a=dg,d^b=dgk,f^a=fi,f^b=fh,g^a=gi,g^b=gi,h^a=hk,h^b=hj,i^a=ik,i^b=ik

Conjugacy classes & character table

218 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226981, #72227126, #72227135
maximal normal subgroup of	#72226981, #72227126, #72227135
maximal quotient of	#72226838

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