#72341559

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting

C₂⁴×C₂²⋊C₄ ⋊ D₃

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂²×#1020
commutator	C₂×C₂²⋊A₄	C₂²×C₄
Fitting	C₂⁴×C₂²⋊C₄	D₃
socle	C₂⁶	C₂²×D₃
prime radical	C₂×#87524	C₂
cosocle	C₂²×C₂²⋊A₄	C₂³
hypercenter	C₂²⋊C₄	#1020
hypocenter	C₂²⋊A₄	C₄×D₄
LCS	G ▷ C₂×C₂²⋊A₄ ▷ C₂²⋊A₄
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

7608 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 7608 conjugacy classes of subgroups — too many to draw. Only its 87 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|b^2=d,c^2=de,b^a=be,f^c=f^2,g^c=gh,g^f=h,h^f=gh,i^c=j,i^f=ij,j^c=i,j^f=i

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341561, #72341604, C₄×D₄×#1020
maximal normal subgroup of	#72341561, #72341604, C₄×D₄×#1020
maximal quotient of	#72341923

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