#72226219

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Splittings

Fitting

C₂×H(7,𝔽₂) ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂	C₂³×GL(2,ℤ/4)
commutator, cosocle	#41⋊C₃	C₂⁵
Frattini	C₄	C₂⁴×S₄
Fitting	C₂×H(7,𝔽₂)	D₃
hypercenter	C₂×2₋¹⁺⁴	S₄
hypocenter	SL(2,𝔽₃)	C₂³×D₄
LCS	G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ C₂×2₋¹⁺⁴

13676 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 96 points:

(1,52,87,26,59,80,9,53,73,5,29,55)(2,40,77,42,83,93,11,78,58,13,49,92)(3,22,96,33,60,70,20,35,62,7,56,28)(4,19,69,24,74,86,8,57,94,21,30,88)(6,85,23,63,48,18,31,95,36,25,68,17)(10,14,90,46,82,72,37,43,54,38,47,66)(12,79,27,41,75,91,34,51,65,44,50,61)(15,84,71,39,45,89,32,81,67,16,76,64);(1,4,9,8)(2,12,11,34)(3,17,20,18)(5,24,26,21)(6,22,31,35)(7,23,33,36)(10,16,37,39)(13,41,42,44)(14,45,43,76)(15,46,32,38)(19,52,57,53)(25,60,63,56)(27,58,65,77)(28,48,70,68)(29,30,59,74)(40,51,78,79)(47,81,82,84)(49,50,83,75)(54,89,90,64)(55,86,80,88)(61,92,91,93)(62,95,96,85)(66,71,72,67)(69,73,94,87);(2,11)(10,37)(12,34)(13,42)(14,43)(15,32)(16,39)(27,65)(38,46)(40,78)(41,44)(45,76)(47,82)(49,83)(50,75)(51,79)(54,90)(58,77)(61,91)(64,89)(66,72)(67,71)(81,84)(92,93);(1,3)(2,10)(4,17)(5,7)(6,19)(8,18)(9,20)(11,37)(12,16)(13,38)(14,40)(15,41)(21,36)(22,52)(23,24)(25,30)(26,33)(27,64)(28,55)(29,56)(31,57)(32,44)(34,39)(35,53)(42,46)(43,78)(45,51)(47,49)(48,86)(50,81)(54,58)(59,60)(61,67)(62,73)(63,74)(65,89)(66,92)(68,88)(69,85)(70,80)(71,91)(72,93)(75,84)(76,79)(77,90)(82,83)(87,96)(94,95);(1,2,4,12,9,11,8,34)(3,16,18,10,20,39,17,37)(5,13,21,44,26,42,24,41)(6,27,22,58,31,65,35,77)(7,32,23,38,33,15,36,46)(14,48,45,70,43,68,76,28)(19,54,53,64,57,90,52,89)(25,61,56,93,63,91,60,92)(29,71,30,72,59,67,74,66)(40,80,79,86,78,55,51,88)(47,85,84,96,82,95,81,62)(49,87,50,69,83,73,75,94)

Polycyclic presentation:

2^6,3,2^3|a^2=f,d^2=j,e^2=j,f^2=j,h^2=j,i^2=j,b^a=bfj,c^a=cj,e^d=ej,f^b=fj,g^a=g^2,h^a=ij,h^g=i,i^a=hj,i^g=hij,i^h=ij

Conjugacy classes & character table

115 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226249
maximal normal subgroup of	#72226249

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.6);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, 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.6*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.9*F.10);;
gap> SetConjugate(coll, 8, 7, F.9);;
gap> SetConjugate(coll, 9, 1, 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);