Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×D32 ⋊ C2 |
| Sylow | C3 ⋊ #2262 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D3×D16 |
| commutator, cosocle | C3×C16 | C23 |
| Frattini | C16 | C22×D3 |
| Fitting | C3×D32 | C2 |
| socle | C2×C3 | C2×D16 |
| hypercenter | D32 | D3 |
| hypocenter | C3 | #2262 |
| LCS | G ▷ C3×C16 ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ D32 | |
106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 67 points:
(2,3)(4,7,10,25)(5,12,16,15)(6,18,22,21)(8,23,29,56)(9,24,32,57)(11,33,35,43)(13,36,40,39)(14,37,42,30)(17,44,45,52)(19,46,49,48)(20,47,51,31)(26,53,59,58)(27,54,61,60)(28,55,62,65)(34,41,63,67)(38,50,64,66);(1,2,3)(4,6)(5,11)(7,18)(8,27)(9,31)(10,22)(12,33)(13,38)(14,41)(15,43)(16,35)(17,26)(19,28)(20,24)(21,25)(23,54)(29,61)(30,34)(32,47)(36,50)(37,63)(39,66)(40,64)(42,67)(44,53)(45,59)(46,55)(48,65)(49,62)(51,57)(52,58)(56,60);(4,5)(6,17)(7,12)(8,26)(9,30)(10,16)(11,19)(13,28)(14,24)(15,25)(18,44)(20,50)(21,52)(22,45)(23,53)(27,41)(29,59)(31,38)(32,37)(33,46)(34,60)(35,49)(36,55)(39,65)(40,62)(42,57)(43,48)(47,64)(51,66)(54,63)(56,58)(61,67)Polycyclic presentation:
2^7,3|c^2=g,d^2=ef,e^2=fg,f^2=g,b^a=bd,d^a=de,e^a=ef,f^a=fg,d^b=de,e^b=ef,f^b=fg,h^c=h^257 conjugacy classes — view the full character table.
| maximal subgroup of | #10716238, #10716240, #10716243, … |
| maximal normal subgroup of | #10716238, #10716240, #10716243, … |
| maximal quotient of | #10715618, #10715687, Dic3×D32, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 1952);