#72226271

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting

#60087 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄×S₄
commutator, cosocle	C₂×#41⋊C₃	C₂⁴
Frattini	C₂×C₄	C₂³×S₄
Fitting	#60087	D₃
hypercenter	#419	S₄
hypocenter	SL(2,𝔽₃)	C₂×#120
LCS	G ▷ C₂×#41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

4612 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 52 points:

(5,25,37,20,39,23,11,38,42,7,22,43)(6,33,47,27,31,49,12,35,51,13,34,50)(8,18,36,10,9,40,21,24,44,19,26,41)(14,45,52,17,28,46,16,29,48,30,15,32);(6,12)(13,27)(14,16)(15,28)(17,30)(29,45)(31,34)(32,46)(33,35)(47,51)(48,52)(49,50);(2,4)(7,10)(8,21)(9,23)(13,17)(14,16)(15,32)(18,36)(19,20)(22,40)(24,44)(25,37)(26,43)(27,30)(28,46)(29,47)(31,49)(33,52)(34,50)(35,48)(38,42)(39,41)(45,51);(1,2,3,4)(5,6)(7,17)(8,16)(9,15)(10,13)(11,12)(14,21)(18,35)(19,27)(20,30)(22,34)(23,32)(24,33)(25,29)(26,28)(31,39)(36,48)(37,47)(38,45)(40,50)(41,49)(42,51)(43,46)(44,52)

Polycyclic presentation:

2^6,3,2^3|a^2=e,d^2=j,f^2=j,h^2=j,i^2=j,b^a=be,c^a=cj,d^a=df,d^b=df,f^a=fj,f^b=fj,f^d=fj,g^b=g^2,h^b=hi,h^g=i,i^b=ij,i^g=hij,i^h=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226288, #72258773
maximal normal subgroup of	#72258773

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