#72242437

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

Aliases

No aliases for this group.

Splittings

Fitting

C₃×#576 ⋊ C₂×D₃

Subgroups

	subgroup	G/H
center	C₂	D₃×D₄×S₄
commutator, cosocle	C₃×#41⋊C₃	C₂⁴
Frattini	C₄	C₂²×D₃×S₄
Fitting	C₃×#576	C₂×D₃
socle	C₂×C₃	C₂×D₄×S₄
hypercenter	Q₁₆	D₃×S₄
hypocenter	C₃×SL(2,𝔽₃)	C₂²×D₄
LCS	G ▷ C₃×#41⋊C₃ ▷ C₃×SL(2,𝔽₃)
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

2785 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 4 generators on 51 points:

(1,35,34,3,36,29,8,23,44,7,17,19)(2,45,46,10,28,37,9,30,48,24,31,41)(4,20,33,16,5,39,15,14,43,6,21,38)(11,27,42,13,26,18,25,47,32,22,12,40)(49,50,51);(3,6)(4,15)(5,19)(7,16)(10,22)(11,25)(12,18)(13,24)(14,33)(17,38)(20,43)(21,29)(23,34)(26,40)(27,46)(28,41)(30,42)(31,37)(32,45)(35,44)(36,39)(47,48);(2,9)(10,24)(11,25)(12,26)(13,22)(18,40)(27,47)(28,31)(30,45)(32,42)(37,41)(46,48)(50,51);(1,2,8,9)(3,10,7,24)(4,11,15,25)(5,18,21,40)(6,22,16,13)(12,29,26,19)(14,32,20,42)(17,37,36,41)(23,46,35,48)(27,44,47,34)(28,38,31,39)(30,33,45,43)

Polycyclic presentation:

2^5,3^2,2^3|a^2=j,d^2=j,e^2=j,h^2=j,i^2=j,b^a=bj,d^c=de,e^c=ej,e^d=ej,f^a=f^2,f^c=f^2,g^b=g^2,h^a=ij,h^c=ij,h^f=hi,i^a=hj,i^c=hj,i^f=h,i^h=ij

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72242441, D₃×#11715001
maximal normal subgroup of	D₃×#11715001

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