#72269082

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

Aliases

No aliases for this group.

Splittings

Fitting

C₃×#568 ⋊ C₂×D₃

Subgroups

	subgroup	G/H
center	C₂	C₂×#10596432
commutator, cosocle	C₃×#41⋊C₃	C₂⁴
Frattini	C₄	C₂²×D₃×S₄
Fitting	C₃×#568	C₂×D₃
socle	C₂×C₃	C₂×D₄×S₄
hypercenter	M₁₆	D₃×S₄
hypocenter	C₃×SL(2,𝔽₃)	C₂²×D₄
LCS	G ▷ C₃×#41⋊C₃ ▷ C₃×SL(2,𝔽₃)
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

2786 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)(4,25,39,10,21,50)(5,32,45)(6,24,48,20,36,23)(7,8,47,9,26,38)(11,51,44)(12,15,37)(13,29,49)(14,30,46)(16,28,43)(17,19,42,18,40,41)(22,31,34)(27,35,33);(5,12)(6,17)(7,9)(8,23)(13,14)(15,33)(16,31)(18,20)(19,38)(21,41)(22,29)(24,50)(25,42)(26,48)(28,37)(30,44)(32,43)(34,46)(35,49)(36,39)(40,47)(45,51);(2,3)(4,6,10,20)(5,11,13,27)(7,17,9,18)(8,19,26,40)(12,16,14,31)(15,28,30,34)(21,36,25,24)(22,37,43,46)(23,39,48,50)(29,35,32,51)(33,45,44,49)(38,41,47,42);(4,5)(6,11)(7,12)(8,22)(9,14)(10,13)(15,23)(16,17)(18,31)(19,37)(20,27)(21,33)(24,49)(25,44)(26,43)(28,39)(29,42)(30,48)(32,41)(34,50)(35,38)(36,45)(40,46)(47,51)

Polycyclic presentation:

2^5,3^2,2^3|b^2=j,e^2=j,h^2=j,i^2=j,c^a=ce,c^b=ce,d^a=dj,e^a=ej,e^b=ej,e^c=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

84 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, 2, F.10);;
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, 5, 3, 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);