#72239366

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#14706
commutator	C₂²×C₄²	C₂³×C₄
Frattini, cosocle	C₂²×C₄×C₈	C₂⁴
socle	C₂²	#393484
prime radical	C₂×#392	M₁₆
LCS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

3739 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,9,11,16)(4,7)(5,8)(6,10,15,14)(12,13)(17,18,21,27,23,28,30,42,24,19,29,32,22,33,25,38)(20,26,35,40,37,41,44,48,34,31,43,45,36,46,39,47);(1,3)(2,6,8,15)(4,10)(5,11)(7,9,12,16)(13,14)(26,31)(40,45)(41,46)(47,48);(1,2)(3,9,11,16)(4,12)(5,8)(6,14,15,10)(7,13)(17,19,30,42,22,28,21,27,24,18,25,38,23,33,29,32)(20,31,44,48,36,41,35,40,34,26,39,47,37,46,43,45);(17,20)(18,26)(19,31)(21,35)(22,36)(23,37)(24,34)(25,39)(27,40)(28,41)(29,43)(30,44)(32,45)(33,46)(38,47)(42,48)

Polycyclic presentation:

2^11|a^2=g,b^2=f,c^2=ghk,e^2=i,g^2=j,h^2=k,j^2=k,b^a=be,c^a=cjk,c^b=cef,d^b=dhj,e^c=ei,e^d=ek,f^a=fi,f^c=fi,g^b=gi,g^c=gk,h^a=hk,h^c=hk

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239490
maximal normal subgroup of	#72239490

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