Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting, Sylow | #3573 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C8 | C22×A4 |
| commutator, hypocenter | Q8 | C22×C3×C4 |
| Frattini | C4 | C23×A4 |
| Fitting | #3573 | C3 |
| socle | C2 | C22×C4×A4 |
| prime radical | #5945 | C2 |
| cosocle | #41 | C23×C3 |
| hypercenter | #131 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C8 ◁ #131 | |
334 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 82 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:
(1,4,8,5)(2,10,14,11)(3,15,18,19)(6,49,7,22,24,23,26,50,27,20,25,21)(9,28,31,32)(12,59,13,35,37,36,39,60,40,33,38,34)(16,61,17,47,43,48,45,62,46,41,44,42)(29,63,30,57,53,58,55,64,56,51,54,52);(1,44)(2,54)(3,24)(4,61)(5,62)(6,46)(7,16)(8,43)(9,37)(10,63)(11,64)(12,56)(13,29)(14,53)(15,50)(17,26)(18,25)(19,49)(20,42)(21,47)(22,48)(23,41)(27,45)(28,60)(30,39)(31,38)(32,59)(33,52)(34,57)(35,58)(36,51)(40,55);(1,2,5,11,8,14,4,10)(3,9,15,28,18,31,19,32)(6,12,22,35,26,39,20,33)(7,13,23,36,27,40,21,34)(16,29,41,51,45,55,47,57)(17,30,42,52,46,56,48,58)(24,37,50,60,25,38,49,59)(43,53,61,63,44,54,62,64)Polycyclic presentation:
2^3,3,2^4|a^2=e,b^2=h,c^2=h,e^2=h,f^2=h,g^2=h,c^b=ch,f^d=fgh,g^d=fh,g^f=gh76 conjugacy classes — view the full character table.
| maximal subgroup of | #33340⋊C3, #60183⋊C3, #60184⋊C3, … |
| maximal normal subgroup of | #33340⋊C3, #60183⋊C3, #60184⋊C3, … |
| maximal quotient of | #60058⋊C3, #60065⋊C3, #60100⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18205);