Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×SD64 ⋊ C2 |
| Sylow | C3 ⋊ #2263 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D3×D16 |
| commutator, cosocle | C3×C16 | C23 |
| Frattini | C16 | C22×D3 |
| Fitting | C3×SD64 | C2 |
| socle | C2×C3 | C2×D16 |
| hypercenter | SD64 | D3 |
| hypocenter | C3 | #2263 |
| LCS | G ▷ C3×C16 ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ SD64 | |
112 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2)(5,11)(10,23)(12,26)(13,27)(14,22)(19,35)(24,28)(25,29);(5,10)(6,15)(7,16)(8,20)(11,23)(12,13)(14,29)(17,34)(18,32)(19,24)(21,33)(22,25)(26,27)(28,35)(30,31);(2,3)(4,5)(6,10)(7,19)(8,22)(9,11)(12,15)(13,16)(14,20)(17,29)(18,23)(21,35)(24,31)(25,30)(26,32)(27,33)(28,34)Polycyclic presentation:
2^7,3|d^2=ef,e^2=fg,f^2=g,b^a=bd,c^a=cg,d^a=de,e^a=ef,f^a=fg,h^a=h^2,d^b=de,e^b=ef,f^b=fg,h^c=h^245 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2263, C2×#69380, #10953330, … |
| maximal normal subgroup of | D3×#2263, C2×#69380, #10953330, … |
| maximal quotient of | #10715625, #10715628, #10715693, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1956);