#72252334

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃²×#99 ⋊ C₂²
Sylow	C₃² ⋊ #12705

Subgroups

	subgroup	G/H
center	C₂	#11784603
commutator	C₂⁴×C₃²	C₂²×C₄
Frattini	C₂²×D₄	C₂×D₃²
Fitting	C₂×C₃²×#99	C₂²
socle	C₂×C₃²	#1781
prime radical	#11904339	C₂
cosocle	C₂²×C₃²×D₄	C₂³
hypercenter	C₂×#99	D₃²
hypocenter	C₃²	#12705
LCS	G ▷ C₂⁴×C₃² ▷ C₂²×C₃² ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×D₄ ◁ C₂×#99

2812 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:

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

Polycyclic presentation:

2^8,3^2|a^2=gh,b^2=dfh,c^2=efh,f^2=h,b^a=bd,c^a=cd,c^b=cde,d^a=dh,d^b=dg,d^c=dgh,e^b=eg,e^c=egh,f^a=fgh,f^b=fg,f^c=fgh,f^d=fh,f^e=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

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252338, #72252386, #72252392
maximal normal subgroup of	#72252338, #72252386, #72252392

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