#72258659

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Splittings

Fitting

C₂×#3585 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄×S₄
commutator, cosocle	C₂×#41⋊C₃	C₂⁴
Frattini	C₂×C₄	C₂³×S₄
Fitting	C₂×#3585	D₃
hypercenter	C₂×#137	S₄
hypocenter	SL(2,𝔽₃)	C₂×#127
LCS	G ▷ C₂×#41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#137

3842 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:

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

Polycyclic presentation:

2^6,3,2^3|a^2=e,c^2=f,f^2=j,h^2=j,i^2=j,b^a=be,c^b=cf,d^a=dj,f^b=fj,g^a=g^2,g^b=g^2,g^c=g^2,h^a=ij,h^b=ij,h^c=ij,h^g=i,i^a=hj,i^b=hj,i^c=hj,i^g=hij,i^h=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#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, 3, F.6);;
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, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 2, F.7^2);;
gap> SetConjugate(coll, 7, 3, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.9*F.10);;
gap> SetConjugate(coll, 8, 2, F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.9*F.10);;
gap> SetConjugate(coll, 8, 7, F.9);;
gap> SetConjugate(coll, 9, 1, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.8*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);