#72245362

order	2304
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆ · Schur multiplier (H₂) = C₃×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₃² ⋊ #7053

Subgroups

	subgroup	G/H
center	C₄²	C₄×C₃⋊Dic₃
commutator	C₃²×C₄	C₄×C₁₆
Frattini	C₈²	C₂×C₃⋊D₃
Fitting	C₃²×#1311	C₂
socle, prime radical	C₂²×C₃²	#322
cosocle	C₃²×C₈²	C₂²
hypercenter	#1311	C₃⋊D₃
hypocenter	C₃²	#7053
LCS	G ▷ C₃²×C₄ ▷ C₃²
UCS	C₁ ◁ C₄² ◁ #1311

366 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 2304

Polycyclic presentation:

2^8,3^2|a^2=c,b^2=d,c^2=e,d^2=f,e^2=g,f^2=h,b^a=bf,c^b=ch,d^a=dh,i^a=i^2,j^a=j^2

Conjugacy classes & character table

408 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72245381, C₃⋊D₃×#7053, #72245454, …
maximal normal subgroup of	#72245381, C₃⋊D₃×#7053, #72245454, …
maximal quotient of	#72245381

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