#72342800

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

Aliases

No aliases for this group.

Splittings

Fitting

#6372744 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₄	C₂²×#1020⋊C₂
commutator, hypocenter	#997	C₂³×C₄
Frattini	C₂²×C₄	C₂³×S₄
Fitting	#6372744	D₃
socle	C₂	C₂×C₄×#1020⋊C₂
prime radical	#72236838	C₂
cosocle	#5950	C₂⁴
hypercenter	M₁₆	#1020⋊C₂
LCS	G ▷ #997
UCS	C₁ ◁ C₄ ◁ M₁₆

5946 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:

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

Polycyclic presentation:

2^5,3,2^5|b^2=e,e^2=k,g^2=k,h^2=k,d^b=dk,d^c=dk,f^c=f^2,g^a=gjk,g^b=gjk,g^c=gj,g^f=h,h^a=hik,h^b=hik,h^c=ghijk,h^f=ghk,h^g=hk,i^c=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, 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, 7, 1, F.7*F.10*F.11);;
gap> SetConjugate(coll, 7, 2, F.7*F.10*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9*F.11);;
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, 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, 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);