#72242452

order	2304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₃

Aliases

No aliases for this group.

Splittings

Fitting

C₃×#3585 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₂	C₂×D₄×C₃⋊S₄
commutator, cosocle	C₃×#41⋊C₃	C₂⁴
Frattini	C₄	C₂³×C₃⋊S₄
Fitting	C₃×#3585	D₃
socle	C₂×C₃	C₂×D₄×S₄
hypercenter	#137	C₃⋊S₄
hypocenter	C₃×SL(2,𝔽₃)	C₂²×D₄
LCS	G ▷ C₃×#41⋊C₃ ▷ C₃×SL(2,𝔽₃)
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

3222 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252919, 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, 4, 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.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 2, F.7^2);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 6, F.9);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
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);