Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
286 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 286 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,23,25)(2,34,37,9,24,58)(3,16,46,14,39,44)(4,19,50,18,47,49)(5,22,7)(8,55,54,27,45,64)(10,11,57,13,32,36)(12,33)(15,59,62)(17,40)(20,48)(21,35)(28,63,30,51,53,31)(29,52)(38,41,61)(43,56);(1,3)(2,8)(4,15)(5,16)(6,17)(7,14)(9,31)(10,27)(11,28)(12,29)(13,30)(18,38)(19,41)(20,42)(21,43)(22,44)(23,39)(24,45)(25,46)(26,40)(32,55)(33,56)(34,51)(35,52)(36,53)(37,54)(47,59)(48,60)(49,61)(50,62)(57,64)(58,63);(1,2,7,13)(3,8,14,30)(4,9,18,10)(5,21,23,12)(6,24,26,11)(15,31,38,27)(16,43,39,29)(17,45,40,28)(19,35,47,33)(20,34,48,32)(22,37,25,36)(41,52,59,56)(42,51,60,55)(44,54,46,53)(49,58,50,57)(61,63,62,64)Polycyclic presentation:
2^4,3,2^3|a^2=h,d^2=h,f^2=h,g^2=h,c^a=cd,d^a=dh,e^a=e^2,f^a=gh,g^a=fh,c^b=ch,d^c=dh,f^e=g,g^e=fgh,g^f=gh31 conjugacy classes — view the full character table.
| maximal subgroup of | #11713431, #11713432, #11713433, … |
| maximal normal subgroup of | #11713431, #11713432, #11713433, … |
| maximal quotient of | #11712098, #11712099, #11712100, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18118);