#72342771

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting

#6372743 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₄	C₂²×#1020⋊C₂
commutator, hypocenter	#997	C₂³×C₄
Frattini	C₂²×C₄	C₂³×S₄
Fitting	#6372743	D₃
socle	C₂	C₂×C₄×#1020⋊C₂
prime radical	#72233471	C₂
cosocle	#5950	C₂⁴
hypercenter	M₁₆	#1020⋊C₂
LCS	G ▷ #997
UCS	C₁ ◁ C₄ ◁ M₁₆

5966 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,42,26,22,7,17,23,16)(2,44,40,37,12,32,38,31)(3,28,47,24,13,6,27,5)(4,55,52,49,20,46,50,45)(8,25,54,21,14,11,41,10)(9,63,60,58,35,57,59,56)(15,39,62,36,29,19,53,18)(30,51,64,48,43,34,61,33);(3,13)(6,26)(8,14)(11,40)(15,29)(16,22)(17,27)(19,52)(23,28)(25,38)(30,43)(31,37)(32,41)(34,60)(39,50)(42,47)(44,54)(45,49)(46,53)(51,59)(55,62)(56,58)(57,61)(63,64);(1,2,4,9,7,12,20,35)(3,14,15,43,13,8,29,30)(5,21,18,48,24,10,36,33)(6,25,19,51,28,11,39,34)(16,31,45,56,22,37,49,58)(17,32,46,57,42,44,55,63)(23,40,50,60,26,38,52,59)(27,41,53,61,47,54,62,64);(3,13)(8,14)(15,29)(16,22)(17,42)(27,47)(30,43)(31,37)(32,44)(41,54)(45,49)(46,55)(53,62)(56,58)(57,63)(61,64)

Polycyclic presentation:

2^5,3,2^5|b^2=e,e^2=k,g^2=k,h^2=k,d^a=dk,d^b=dk,d^c=dk,f^c=f^2,f^d=f^2,g^b=gjk,g^c=gj,g^d=gk,g^f=h,h^b=hik,h^c=ghijk,h^d=gh,h^f=ghk,h^g=hk,i^c=ij,i^d=ij,i^f=ij,i^g=ik,j^f=i,j^h=jk

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342856
maximal normal subgroup of	#72342856

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