Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×#368 ⋊ C2 |
| Sylow | C3 ⋊ #2222 |
140 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 45 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 — 3 generators on 67 points:
(1,2)(4,7)(5,13)(6,20)(8,26)(9,32)(10,27)(11,28)(12,34)(14,38)(15,42)(16,25)(17,39)(18,43)(19,48)(21,50)(22,54)(23,51)(24,52)(29,56)(30,59)(31,58)(33,60)(35,63)(36,55)(37,62)(40,64)(41,57)(44,66)(45,49)(46,65)(47,67)(53,61);(4,6,19,47,10,23,31,33,11,24,30,53,9,22,8,21)(5,12,29,45,16,36,41,46,17,37,40,44,15,35,14,18)(7,20,48,67,27,51,58,60,28,52,59,61,32,54,26,50)(13,34,56,49,25,55,57,65,39,62,64,66,42,63,38,43);(2,3)(4,5,11,17)(6,18,24,46)(7,25,28,42)(8,29,31,40)(9,16,10,15)(12,33,37,21)(13,32,39,27)(14,30,41,19)(20,49,52,66)(22,45,23,44)(26,57,58,38)(34,61,62,67)(35,53,36,47)(43,54,65,51)(48,56,59,64)(50,55,60,63)Polycyclic presentation:
2^7,3|a^2=g,b^2=df,d^2=fg,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fg,h^a=h^2,e^c=eg,h^c=h^248 conjugacy classes — view the full character table.
| maximal subgroup of | #10715886, #10715888, #10715893, … |
| maximal normal subgroup of | #10715886, #10715888, #10715893, … |
| maximal quotient of | #10709399, #10709416, #10709450, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1938);