#72252354

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Splittings

Fitting	C₃²×#392 ⋊ C₂²
Sylow	C₃² ⋊ #18162

Subgroups

	subgroup	G/H
center	C₂²	#10595278
commutator, cosocle	C₂²×C₃²×C₄	C₂⁴
Frattini	C₂²×C₄	C₂²×D₃²
Fitting	C₃²×#392	C₂²
socle	C₂²×C₃²	C₂×#120
hypercenter	#392	D₃²
hypocenter	C₃²	#18162
LCS	G ▷ C₂²×C₃²×C₄ ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #392

4314 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 22 points:

(10,14)(13,15)(18,19)(20,21);(1,3)(2,5)(4,7)(6,8)(10,14)(13,15)(17,18,19);(5,8)(10,13)(11,16)(14,15);(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,e^2=h,b^a=be,c^a=ch,c^b=cf,d^a=dh,e^a=eh,e^b=eh,e^c=eg,f^a=fg,i^a=i^2,i^d=i^2,j^a=i^2j,j^d=j^2

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252358
maximal normal subgroup of	#72252358

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