Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C3×#390 ⋊ C2 |
| Sylow | C3 ⋊ #2699 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#390 | C2 |
| hypercenter | #390 | D3 |
| hypocenter | C3 | #2699 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #390 | |
612 conjugacy classes of subgroups; only the 137 normal subgroups are shown.
This group has 612 conjugacy classes of subgroups — too many to draw. Only its 137 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(9,13)(12,15)(17,20)(21,23);(2,3)(5,7)(9,13)(12,15)(16,18,19,22)(17,21,20,23);(2,3)(4,5)(6,7)(8,10,11,14)(9,12,13,15)(16,17,19,20)(18,23,22,21);(8,9,11,13)(10,12,14,15)(17,20)(18,22)Polycyclic presentation:
2^7,3|a^2=e,b^2=f,c^2=ef,b^a=bef,c^a=cf,d^a=de,c^b=cg,d^b=def,h^b=h^2,h^c=h^260 conjugacy classes — view the full character table.
| maximal subgroup of | #10707266, #10707270, #10707954, … |
| maximal normal subgroup of | #10707266, #10707270, #10707954, … |
| maximal quotient of | #10722548, #10722599, #10722704, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 8961);