#72252317

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Sylow

C₃² ⋊ #8346

Subgroups

	subgroup	G/H
center	C₂²	#10591706
commutator	C₂²×C₃²×C₄	C₂²×C₄
Frattini	C₂³×C₄	C₂×D₃²
Fitting	C₂²×C₃²×C₂²⋊C₄	C₂²
socle	C₂²×C₃²	#379
prime radical	C₃×#85697	C₂
cosocle	C₂³×C₃²×C₄	C₂³
hypercenter	C₂²×C₂²⋊C₄	D₃²
hypocenter	C₃²	#8346
LCS	G ▷ C₂²×C₃²×C₄ ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₂²⋊C₄

2518 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 3 generators on 22 points:

(1,3)(2,5)(4,7)(6,8)(17,18,19)(20,21,22);(5,8)(10,13,14,15)(18,19)(21,22);(1,2,4,6)(3,5,7,8)(9,10)(11,13)(12,14)(15,16)(21,22)

Polycyclic presentation:

2^8,3^2|a^2=g,b^2=d,f^2=h,b^a=bf,c^b=ce,d^a=dh,e^a=eg,f^a=fh,f^c=fg,i^a=i^2,i^b=i^2,j^a=i^2j,j^b=j^2

Conjugacy classes & character table

162 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252321, #72252335
maximal normal subgroup of	#72252321, #72252335

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