#72215293

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting

C₂⁶ ⋊ D₃²

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×S₄²
commutator, cosocle	C₂×A₄²	C₂³
Fitting	C₂⁶	D₃²
socle	C₂⁵	C₂×D₃²
hypercenter	C₂²	S₄²
hypocenter	A₄²	C₂×D₄
LCS	G ▷ C₂×A₄² ▷ A₄²
UCS	C₁ ◁ C₂ ◁ C₂²

2706 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 2706 conjugacy classes of subgroups — too many to draw. Only its 55 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:

2^4,3^2,2^4|a^2=d,b^a=bd,e^a=e^2,e^c=e^2,f^a=e^2f,f^c=f^2,g^e=gh,g^f=h,h^a=gh,h^c=gh,h^e=g,h^f=gh,i^a=gi,i^c=ghj,i^e=gi,i^f=j,j^c=hi,j^e=hj,j^f=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72228310, C₂×#72215293, C₃×#72215293, …
maximal normal subgroup of	#72228310, C₂×#72215293, C₃×#72215293, …
maximal quotient of	#72226218, #72226221, #72226236, …
holomorph of	SL(2,ℤ/4), C₂×S₄

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