Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #67607 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C3×D32 ⋊ C2 |
| Sylow | C3 ⋊ D64 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #4519 |
| commutator, cosocle | C3×C32 | C22 |
| Frattini | C32 | C2×D3 |
| Fitting | C3×D32 | C2 |
| socle | C2×C3 | D32 |
| hypercenter | D32 | D3 |
| hypocenter | C3 | D64 |
| LCS | G ▷ C3×C32 ▷ C3×C16 ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ D32 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 2 generators on 67 points:
(1,2,3)(5,11)(6,17)(7,18)(8,23)(9,27)(12,13)(14,39)(15,41)(16,33)(19,49)(20,52)(21,45)(22,30)(24,58)(25,48)(26,31)(28,56)(29,32)(34,54)(35,53)(36,38)(37,60)(40,64)(42,43)(44,50)(46,47)(51,67)(55,62)(57,61)(59,63)(65,66);(2,3)(4,5)(6,11)(7,22)(8,26)(9,29)(10,16)(12,17)(13,18)(14,23)(15,27)(19,39)(20,41)(21,33)(24,57)(25,55)(28,59)(30,43)(31,42)(32,50)(34,49)(35,52)(36,45)(37,58)(38,48)(40,56)(44,60)(46,54)(47,53)(51,64)(61,67)(62,66)(63,65)Polycyclic presentation:
2^7,3|c^2=de,d^2=ef,e^2=fg,f^2=g,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,h^a=h^2,c^b=cd,d^b=de,e^b=ef,f^b=fg54 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D64, C2×#67607, #10716239, … |
| maximal normal subgroup of | D3×D64, C2×#67607, #10716239, … |
| maximal quotient of | #10683064, #10683065, #10683086, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 183);