#72252290

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃²×H(5,𝔽₂) ⋊ C₂²
Sylow	C₃² ⋊ #23897

Subgroups

	subgroup	G/H
center	C₂	#11809009
commutator, cosocle	C₂²×C₃²×C₄	C₂⁴
Frattini	C₂²×C₄	C₂²×D₃²
Fitting	C₂×C₃²×H(5,𝔽₂)	C₂²
socle	C₂×C₃²	#2403
hypercenter	C₂×H(5,𝔽₂)	D₃²
hypocenter	C₃²	#23897
LCS	G ▷ C₂²×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×H(5,𝔽₂)

4854 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:

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

Polycyclic presentation:

2^8,3^2|a^2=g,c^2=e,f^2=h,b^a=bf,c^a=cf,c^b=ce,d^a=dg,e^a=eh,f^b=fh,f^d=fh,g^b=gh,g^c=gh,i^a=i^2,i^c=i^2,j^a=i^2j,j^c=j^2

Conjugacy classes & character table

132 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252304, #72252386
maximal normal subgroup of	#72252304, #72252386

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