Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Fitting | C3×#322 ⋊ C2 |
| Sylow | C3 ⋊ #1454 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C42 | C22×D3 |
| commutator, socle | C22×C3 | C2×C42 |
| Fitting | C3×#322 | C2 |
| prime radical | C22×D3 | C42 |
| cosocle | C3×C42 | C23 |
| hypercenter | #322 | D3 |
| hypocenter | C3 | #1454 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C42 ◁ #322 | |
178 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 178 conjugacy classes of subgroups — too many to draw. Only its 89 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) = 192 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2)(5,10)(6,12)(11,17)(13,18)(21,26)(25,31)(27,33)(32,35);(4,6)(5,9)(7,13)(8,12)(10,16)(11,15)(14,18)(17,19)(20,22,23,28,24,29,30,34)(21,25,27,32,26,31,33,35);(2,3)(4,5,7,11,8,10,14,17)(6,9,13,15,12,16,18,19)(20,21)(22,25)(23,27)(24,26)(28,32)(29,31)(30,33)(34,35)Polycyclic presentation:
2^7,3|a^2=d,b^2=e,d^2=f,e^2=g,c^a=cfg,h^a=h^2,c^b=cf,h^c=h^296 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1454, C2×#68074, #10760819, … |
| maximal normal subgroup of | D3×#1454, C2×#68074, #10760819, … |
| maximal quotient of | #10684561, #10684577, #10685621, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 650);