#72342728

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

Aliases

No aliases for this group.

Splittings

Fitting

#6372742 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₈	C₂×#1020⋊C₂
commutator, hypocenter	#997	C₂³×C₄
Frattini	C₂²×C₄	C₂³×S₄
Fitting	#6372742	D₃
socle	C₂	C₂×C₄×#1020⋊C₂
prime radical	#72233471	C₂
cosocle	#5950	C₂⁴
hypercenter	C₂×C₈	#1020⋊C₂
LCS	G ▷ #997
UCS	C₁ ◁ C₈ ◁ C₂×C₈

5992 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342856, #72342925
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, 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, 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, 3, F.7*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, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> SetConjugate(coll, 9, 6, F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.11);;
gap> SetConjugate(coll, 9, 8, F.9*F.11);;
gap> SetConjugate(coll, 10, 6, F.9*F.10);;
gap> SetConjugate(coll, 10, 8, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);