#72335137

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 6949 conjugacy classes of subgroups; only the 42288 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 20 points:

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

Polycyclic presentation:

2^11|g^2=i,b^a=bh,c^b=ck,d^a=dk,d^c=dk,g^a=ghk,g^b=gi,g^c=gj,g^d=gij,g^e=ghk,g^f=gi

Conjugacy classes & character table

320 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72335137
maximal normal subgroup of	C₂×#72335137
maximal quotient of	#72335164, C₂×#72335137

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