Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Sylow | C3 ⋊ #2642 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, prime radical, cosocle | C23×C3 | C24 |
| Fitting | C3×#376 | C2 |
| hypercenter | #376 | D3 |
| hypocenter | C3 | #2642 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #376 | |
416 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 416 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3)(16,19,20,27)(17,23,21,29)(18,24,25,26)(22,30,28,31);(2,3)(4,5)(6,7)(9,13)(12,15)(16,18,20,25)(17,22,21,28)(19,26,27,24)(23,31,29,30);(4,5,6,7)(8,10,11,14)(9,12,13,15)(17,21)(22,28)(23,29)(30,31);(4,5,6,7)(8,9)(10,12)(11,13)(14,15)(16,17,20,21)(18,22,25,28)(19,23,27,29)(24,30,26,31)Polycyclic presentation:
2^7,3|a^2=e,b^2=f,c^2=eg,d^2=eg,b^a=beg,c^a=cf,c^b=cg,d^c=deg,h^c=h^272 conjugacy classes — view the full character table.
| maximal subgroup of | #10696577, #10696581, #10696633, … |
| maximal normal subgroup of | #10696577, #10696581, #10696633, … |
| maximal quotient of | #10721589, #10721719, #10721791, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5983);