#72210723

order	5376
automorphism	?
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting, socle, Sylow	C₂⁸ ⋊ ΓL(1,𝔽₈)
derived, Hall	C₂²×#90632 ⋊ C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₂²×#90632	C₃
Fitting, socle	C₂⁸	ΓL(1,𝔽₈)
LCS	G ▷ C₂²×#90632
UCS	C₁

20259 conjugacy classes of subgroups; only the 11 normal subgroups are shown.

normal subgroups only

This group has 20259 conjugacy classes of subgroups — too many to draw. Only its 11 normal subgroups are shown.

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

Full subgroup data →

Presentation

Polycyclic presentation:

3,2^2,7,2^6|b^a=c,c^a=bc,d^a=d^4,e^a=j,e^d=efj,f^a=fgh,f^d=fgh,g^a=fj,g^d=fg,h^a=ehj,h^d=fgi,i^a=efhi,i^d=ei,j^a=efghij,j^d=efgj

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

SL(3,ℤ/4)

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