Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #3585⋊C3 |
| Fitting, Sylow | #3585 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×D4×A4 |
| commutator, cosocle | #41 | C23×C3 |
| Frattini | C4 | C23×A4 |
| Fitting | #3585 | C3 |
| hypercenter | #137 | A4 |
| hypocenter | Q8 | C2×C3×D4 |
| LCS | G ▷ #41 ▷ Q8 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #137 | |
339 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 339 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,4,19)(2,26,27,9,10,43)(3,15,34)(5,20,36)(6,24,41)(7,23,38)(8,31,42,12,25,47)(11,32,44,13,28,48)(14,29,45,30,33,46)(16,35,37)(17,21,39)(18,22,40);(2,13)(4,20)(5,17)(6,16)(8,14)(9,11)(10,26)(12,30)(15,24)(19,41)(21,23)(22,35)(25,31)(27,45)(34,39)(36,40)(37,38)(42,44)(43,46)(47,48);(1,2,7,9)(3,14,18,30)(4,10,23,26)(5,13,17,11)(6,12,16,8)(15,33,22,29)(19,27,38,43)(20,32,21,28)(24,31,35,25)(34,45,40,46)(36,48,39,44)(37,42,41,47)Polycyclic presentation:
2^3,3,2^4|a^2=h,b^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=ch,e^a=eh,e^b=eh,f^d=fgh,g^d=fh,g^f=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | #60388⋊C3, #60390⋊C3, #60470⋊C3, … |
| maximal normal subgroup of | #60388⋊C3, #60390⋊C3, #60470⋊C3, … |
| maximal quotient of | #60087⋊C3, #60089⋊C3, #60109⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18216);