#72225218

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#1619
commutator	C₈²	C₂³×C₄
Frattini, cosocle	C₂×C₈²	C₂⁴
prime radical	C₂×#123275	C₂
LCS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₄² ▷ C₂² ▷ C₁

3658 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^11|d^2=g,e^2=hj,f^2=ik,h^2=j,i^2=k,c^a=cf,c^b=ce,d^c=dj,e^a=ei,e^b=eh,e^c=eh,f^a=fi,f^b=fi,f^c=fi,h^a=hk,h^b=hj,h^c=hj,i^a=ik,i^b=ik,i^c=ik

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225289, #72226981
maximal normal subgroup of	#72225289, #72226981
maximal quotient of	#72225219, #72226961, #72227164

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