#72342747

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	#72236922	C₂
cosocle	#5950	C₂⁴
hypercenter	M₁₆	#1020⋊C₂
LCS	G ▷ #997
UCS	C₁ ◁ C₄ ◁ M₁₆

5942 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:

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

Polycyclic presentation:

2^5,3,2^5|b^2=e,c^2=k,e^2=k,g^2=k,h^2=k,c^a=ck,c^b=ck,f^d=f^2,g^b=gjk,g^c=gjk,g^d=gk,g^f=h,h^b=hik,h^c=hik,h^d=gh,h^f=ghk,h^g=hk,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, 3, F.11);;
gap> SetPower(coll, 5, F.11);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.11);;
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*F.11);;
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.8*F.9*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, 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);