Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C3 ⋊ #2931 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D3 |
| commutator, socle | C22×C3 | C23×C4 |
| Fitting, prime radical | C3×#532 | C2 |
| cosocle | C2×C3×C4 | C24 |
| hypercenter | #532 | D3 |
| hypocenter | C3 | #2931 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ #532 | |
500 conjugacy classes of subgroups; only the 229 normal subgroups are shown.
This group has 500 conjugacy classes of subgroups — too many to draw. Only its 229 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)(8,11)(9,15)(10,17)(12,20)(13,19)(14,21)(16,22)(18,23);(2,3)(8,10,12,18,13,9,14,16)(11,17,20,23,19,15,21,22);(5,7)(11,19)(15,17)(20,21)(22,23);(2,3)(4,5)(6,7)(8,9,12,16,13,10,14,18)(11,17,20,23,19,15,21,22)Polycyclic presentation:
2^7,3|a^2=f,c^2=f,f^2=g,b^a=be,d^a=dg,h^a=h^2,d^b=dg,h^c=h^284 conjugacy classes — view the full character table.
| maximal subgroup of | #10686192, #10686193, #10686202, … |
| maximal normal subgroup of | #10686192, #10686193, #10686202, … |
| maximal quotient of | #10765703, #10765711, #10765719, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 15851);