Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×C3×Q32 ⋊ C2 |
| Sylow | C3 ⋊ #3417 |
350 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 67 points:
(1,2,3)(7,21)(12,35)(17,45)(22,49)(23,50)(32,58)(36,60)(37,54)(43,65)(46,57)(47,55)(51,64)(52,67)(53,61)(59,63)(62,66);(2,3)(4,7)(5,12)(6,17)(8,22)(9,23)(10,21)(11,32)(13,36)(14,37)(15,35)(16,43)(18,46)(19,47)(20,45)(24,51)(25,52)(26,50)(27,53)(28,49)(29,54)(30,55)(31,57)(33,59)(34,58)(38,61)(39,62)(40,60)(41,63)(42,64)(44,65)(48,66)(56,67);(4,6,10,20)(5,11,15,34)(7,17,21,45)(8,25,28,56)(9,30,26,19)(12,32,35,58)(13,39,40,48)(14,41,29,33)(16,42,44,24)(18,38,31,27)(22,52,49,67)(23,55,50,47)(36,62,60,66)(37,63,54,59)(43,64,65,51)(46,61,57,53);(4,5,10,15)(6,16,20,44)(7,12,21,35)(8,24,28,42)(9,29,26,14)(11,31,34,18)(13,38,40,27)(17,43,45,65)(19,48,30,39)(22,51,49,64)(23,54,50,37)(25,41,56,33)(32,57,58,46)(36,61,60,53)(47,66,55,62)(52,63,67,59)Polycyclic presentation:
2^7,3|a^2=g,b^2=g,e^2=fg,f^2=g,b^a=be,e^a=ef,f^a=fg,e^b=ef,f^b=fg,d^c=dg,h^c=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10715877, #10715881, #10715891, … |
| maximal normal subgroup of | #10715877, #10715881, #10715891, … |
| maximal quotient of | #10948101, #10948227, #10948534, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 14604);