#72252398

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

Aliases

No aliases for this group.

Splittings

Fitting	C₃²×B(4,𝔽₂) ⋊ C₂²
Sylow	C₃² ⋊ #33551

Subgroups

	subgroup	G/H
center	C₂	#11865254
commutator, cosocle	C₂×C₃²×D₄	C₂⁴
Frattini	C₂×D₄	C₂²×D₃²
Fitting	C₃²×B(4,𝔽₂)	C₂²
socle	C₂×C₃²	C₂×B(4,𝔽₂)
hypercenter	B(4,𝔽₂)	D₃²
hypocenter	C₃²	#33551
LCS	G ▷ C₂×C₃²×D₄ ▷ C₂²×C₃² ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ B(4,𝔽₂)

3161 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:

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

Polycyclic presentation:

2^8,3^2|a^2=gh,e^2=h,b^a=be,c^a=cgh,c^b=cef,d^a=dgh,e^b=eh,e^c=eg,e^d=eh,f^a=fg,f^b=fh,f^c=fg,f^d=fh,f^e=fh,g^b=gh,i^a=i^2,i^d=i^2,j^a=i^2j,j^d=j^2

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252400
maximal normal subgroup of	#72252400

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*F.8);;
gap> SetPower(coll, 5, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
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);