Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #67607 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C3 ⋊ SD128 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #4519 |
| commutator, cosocle | C3×C32 | C22 |
| Frattini | C32 | C2×D3 |
| Fitting, prime radical | C3×D32 | C2 |
| socle | C2×C3 | D32 |
| hypercenter | D32 | D3 |
| hypocenter | C3 | SD128 |
| LCS | G ▷ C3×C32 ▷ C3×C16 ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ D32 | |
38 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)(14,42)(15,44)(16,33)(19,50)(20,53)(21,34)(22,30)(24,58)(25,38)(26,31)(28,41)(29,32)(35,55)(36,54)(37,40)(39,60)(43,67)(45,46)(47,51)(48,49)(52,66)(56,63)(57,62)(59,65)(61,64);(2,3)(4,5,10,16)(6,11,21,33)(7,22,25,56)(8,26,28,59)(9,29,27,15)(12,34,37,17)(13,38,40,18)(14,41,43,23)(19,42,52,67)(20,44,51,32)(24,57,48,55)(30,61,63,46)(31,64,65,45)(35,66,62,50)(36,47,60,53)(39,49,54,58)Polycyclic presentation:
2^7,3|a^2=g,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×SD128, C2×#67608, #10716238, … |
| maximal normal subgroup of | D3×SD128, C2×#67608, #10716238, … |
| maximal quotient of | #10683065, #10683067, #10683088, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 184);