Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting, Sylow | #72210036 ⋊ C3 |
| derived | #8093479 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #35346⋊C3 |
| commutator, cosocle, hypocenter | #8093479 | C2×C3 |
| Frattini | C25 | C2×C22⋊A4 |
| Fitting | #72210036 | C3 |
| hypercenter | C23 | #85645 |
| LCS | G ▷ #8093479 | |
| UCS | C1 ◁ C22 ◁ C23 | |
4758 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 4758 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 72 points:
(2,3,5)(6,7,8)(9,10)(11,17,25,16,12,36)(13,40,14,18,29,19)(15,20)(21,35,28,32,24,39)(22,60,53,33,48,64)(23,37,54,34,26,65)(27,61,49,38,52,62)(30,67,31,42,57,45)(41,71,43,55,68,51)(44,46,66,56,58,70)(47,63,72,59,50,69);(2,6)(3,7)(9,29)(10,40)(11,49)(12,48)(13,14)(15,57)(16,62)(17,60)(18,19)(20,67)(21,71)(22,23)(24,70)(25,54)(26,27)(28,72)(30,31)(32,68)(33,34)(35,66)(36,65)(37,38)(39,69)(41,44)(42,45)(43,63)(46,47)(50,51)(52,53)(55,56)(58,59)(61,64);(1,5)(2,3)(4,8)(6,7)(9,21,15,25)(10,32,20,36)(11,28,24,12)(13,46,30,52)(14,47,31,53)(16,39,35,17)(18,61,42,58)(19,59,45,64)(22,55,50,26)(23,56,51,27)(29,71,57,54)(33,37,63,41)(34,44,43,38)(40,65,67,68)(48,49,72,70)(60,66,69,62)Polycyclic presentation:
2,3,2^9|c^2=k,d^2=k,c^a=cg,c^b=cd,d^a=dh,d^b=c,d^c=dk,e^b=f,e^c=ei,e^d=ej,f^b=ef,f^c=fj,f^d=fij,g^b=gh,g^e=gk,h^b=g,h^f=hk,i^a=ik54 conjugacy classes — view the full character table.
| maximal subgroup of | #72227737 |
| maximal normal subgroup of | #72227737 |
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, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.11);;
gap> SetPower(coll, 4, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.3);;
gap> SetConjugate(coll, 4, 3, F.4*F.11);;
gap> SetConjugate(coll, 5, 2, F.6);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.5*F.6);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.11);;
gap> SetConjugate(coll, 8, 2, F.7);;
gap> SetConjugate(coll, 8, 6, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);