#72226264

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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×#121
LCS	G ▷ C₂×#41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#137

3411 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:

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

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^b=df,f^b=fj,f^d=fj,g^a=g^2,g^b=g^2,h^a=ij,h^b=ij,h^g=i,i^a=hj,i^b=hj,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, 2, F.4*F.6);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 2, 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, 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, 7, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 8, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);