#72258766

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting

C₂⁴×#41 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₂	C₂×#87524
commutator, cosocle	C₄×C₂²⋊A₄	C₂³
Frattini	C₄	C₂²×#1020
Fitting	C₂⁴×#41	D₃
socle	C₂⁵	C₂×#67
hypercenter	#41	#1020
hypocenter	C₂²⋊A₄	AGL(1,ℤ/8)
LCS	G ▷ C₄×C₂²⋊A₄ ▷ C₂×C₂²⋊A₄ ▷ C₂²⋊A₄
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #41

7947 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 7947 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.

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

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 48

Polycyclic presentation:

2^5,3,2^4|a^2=e,d^2=e,b^a=bd,c^a=ce,d^a=de,d^b=de,f^b=f^2,g^f=h,h^b=gh,h^f=gh,i^b=hij,i^f=j,j^b=gj,j^f=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,ℤ/8)×#1020, #72258962
maximal normal subgroup of	AGL(1,ℤ/8)×#1020

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