Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#385 ⋊ C2 |
| Sylow | C3 ⋊ #2826 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, cosocle | C23×C3 | C24 |
| Fitting | C3×#385 | C2 |
| prime radical | #4960 | C2 |
| hypercenter | #385 | D3 |
| hypocenter | C3 | #2826 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #385 | |
464 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 464 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(6,10)(8,11)(20,22)(21,25)(23,27)(24,26);(2,3)(5,9)(8,11)(14,18)(16,19)(22,27)(24,25);(4,6)(5,8)(7,10)(9,11)(12,14)(13,16)(15,18)(17,19)(20,22,23,27)(21,24,26,25);(4,5)(6,8)(7,9)(10,11)(12,13,15,17)(14,16,18,19)(20,21,23,26)(22,25,27,24)Polycyclic presentation:
2^7,3|a^2=g,b^2=e,b^a=be,c^a=cf,c^b=cg,d^b=def,d^c=de,h^c=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10707488, #10707492, #10708074, … |
| maximal normal subgroup of | #10707488, #10707492, #10708074, … |
| maximal quotient of | #10725824, #10725830, #10725980, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 7228);