#72258754

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

solvable

Aliases

No aliases for this group.

Splittings

Fitting

#62811 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×GL(2,ℤ/4)
commutator, cosocle	C₂×#41⋊C₃	C₂⁴
Frattini	C₂×C₄	C₂³×S₄
Fitting	#62811	D₃
socle	C₂²	C₂²×GL(2,ℤ/4)
hypercenter	#525	S₄
hypocenter	SL(2,𝔽₃)	C₂×#121
LCS	G ▷ C₂×#41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #525

3544 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:

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

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^a=cf,d^a=dj,f^a=fj,g^c=g^2,h^c=ij,h^g=i,i^c=hj,i^g=hij,i^h=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258773, #72258787
maximal normal subgroup of	#72258773, #72258787

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