GL(2,ℤ/8) #72210852

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

Aliases

general linear

GL(2,ℤ/8)

Splittings

Fitting

C₂²×#392 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₂²	C₂×#857⋊C₂
commutator, cosocle	C₂×#95⋊C₃	C₂³
Frattini, socle	C₂⁴	C₂²×S₄
Fitting	C₂²×#392	D₃
hypercenter	C₂³	#857⋊C₂
hypocenter	#95⋊C₃	C₂×D₄
LCS	G ▷ C₂×#95⋊C₃ ▷ #95⋊C₃
UCS	C₁ ◁ C₂² ◁ C₂³

2265 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 2265 conjugacy classes of subgroups — too many to draw. Only its 46 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:

2^4,3,2^5|a^2=d,b^2=j,f^2=h,g^2=i,b^a=bdj,c^a=ci,e^a=e^2,e^c=eij,f^a=fg,f^b=fhj,f^e=ghi,g^a=gi,g^b=gij,g^e=fgh,g^f=gj,h^a=hij,h^e=i,i^e=hij

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×GL(2,ℤ/8), #72258516, #72260440, …
maximal normal subgroup of	C₂×GL(2,ℤ/8), #72258516
maximal quotient of	#72218286, #72218337, #72228212, …
automorphism group of	C₈²
outer automorphism group of	C₈², #1273

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