#72226319

order	4608
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂⁴×C₃

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#10586066 ⋊ C₃²

Subgroups

	subgroup	G/H
center, Frattini, socle	C₂²	C₂³×A₄²
commutator, cosocle	C₂×H(5,𝔽₂)	C₂³×C₃²
Fitting	#10586066	C₃²
hypercenter	#120	A₄²
hypocenter	H(5,𝔽₂)	C₂×C₃²×D₄
LCS	G ▷ C₂×H(5,𝔽₂) ▷ H(5,𝔽₂)
UCS	C₁ ◁ C₂² ◁ #120

9104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 52 points:

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

Polycyclic presentation:

2^3,3^2,2^6|a^2=f,g^2=k,h^2=k,b^a=bf,c^a=ck,g^d=gh,g^e=gh,h^d=g,h^e=g,h^g=hk,i^d=j,i^e=gik,i^g=ik,i^h=ik,j^d=ij,j^e=ghjk,j^g=jk

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.8);;
gap> SetConjugate(coll, 8, 4, F.7);;
gap> SetConjugate(coll, 8, 5, F.7);;
gap> SetConjugate(coll, 8, 7, F.8*F.11);;
gap> SetConjugate(coll, 9, 4, F.10);;
gap> SetConjugate(coll, 9, 5, F.7*F.9*F.11);;
gap> SetConjugate(coll, 9, 7, F.9*F.11);;
gap> SetConjugate(coll, 9, 8, F.9*F.11);;
gap> SetConjugate(coll, 10, 4, F.9*F.10);;
gap> SetConjugate(coll, 10, 5, F.7*F.8*F.10*F.11);;
gap> SetConjugate(coll, 10, 7, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);