#72269088

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂	C₂×#10596473
commutator, cosocle	C₃×#41⋊C₃	C₂⁴
Frattini	C₄	C₂²×D₃×S₄
Fitting	C₃×#585	C₂×D₃
socle	C₂×C₃	C₂×D₄×S₄
hypercenter	#41	D₃×S₄
hypocenter	C₃×SL(2,𝔽₃)	C₂²×D₄
LCS	G ▷ C₃×#41⋊C₃ ▷ C₃×SL(2,𝔽₃)
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #41

2878 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

87 conjugacy classes — view the full character table.

Relationships

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