#72234537

order	8192
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂³⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁶	C₂⁷
LCS	G ▷ C₂⁶ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁶ ▷ C₁

244096336481 conjugacy classes of subgroups; only the 2998862 normal subgroups are shown.

normal subgroups only

This group has 244096336481 conjugacy classes of subgroups — too many to draw. Only its 2998862 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^13|g^2=h,g^a=gm,g^b=gi,g^c=gl,g^d=gk,g^e=gh,g^f=gj

Conjugacy classes & character table

2144 conjugacy classes. The character table is too large to display.

Relationships

automorphism group of

#1436, #2325, #2329, …

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", "l", "m");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.13);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.12);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 7, 5, F.7*F.8);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);