#72215118

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#72994
commutator	C₄×C₄²⋊C₃	C₂×C₄
Frattini	C₂³×C₄	C₂×S₄
Fitting	C₂×#1842	D₃
socle	C₂³	C₄⋊SL(2,ℤ/4)
prime radical	C₂²×C₄²⋊C₃	Q₈
cosocle	C₂×C₄×C₄²⋊C₃	C₂²
hypercenter	C₂×C₄	#857⋊C₂
hypocenter	C₄²⋊C₃	#108
LCS	G ▷ C₄×C₄²⋊C₃ ▷ C₂×C₄²⋊C₃ ▷ C₄²⋊C₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄

405 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 405 conjugacy classes of subgroups — too many to draw. Only its 36 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:

2^5,3,2^4|a^2=c,b^2=dej,c^2=e,d^2=e,g^2=i,h^2=j,b^a=bd,d^a=de,f^a=f^2,f^b=fij,f^d=fi,g^a=ghi,g^b=gi,g^f=hi,h^b=hj,h^f=gh,i^a=ij,i^f=j,j^f=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215201, #72218214
maximal normal subgroup of	#72218214
maximal quotient of	#72215126

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