#72232064

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁷
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 12901 conjugacy classes of subgroups; only the 39256 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 24 points:

(11,15)(14,16);(11,15)(14,16)(17,18)(19,23)(20,24)(21,22);(11,15)(14,16)(18,22)(20,23);(2,4)(6,8)(10,13)(14,16)(19,24)(20,23);(10,13)(14,16)(19,24)(20,23);(5,6)(7,8)(9,10,12,13)(11,14,15,16)(17,19,21,24)(18,23,22,20);(1,2)(3,4)(5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,20,21,23)(18,24,22,19)

Polycyclic presentation:

2^11|f^2=i,g^2=h,c^b=ch,f^d=fk,f^e=fi,g^a=ghi,g^b=ghi,g^c=gi,g^d=gj,g^e=gh

Conjugacy classes & character table

296 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232064
maximal normal subgroup of	C₂×#72232064
maximal quotient of	#72323772, C₂×#72232064, #72336510
automorphism group of	#1478, #1479

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