#72269070

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂	C₂×#67×S₄
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

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

Polycyclic presentation:

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

96 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, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*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);