#72342787

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	#72236922	C₂
cosocle	#5950	C₂⁴
hypercenter	C₂×C₈	#1020⋊C₂
LCS	G ▷ #997
UCS	C₁ ◁ C₈ ◁ C₂×C₈

5944 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:

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