Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| Sylow | C3 ⋊ #2844 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D3 |
| commutator, socle, prime radical, cosocle | C23×C3 | C24 |
| Fitting | C3×#382 | C2 |
| hypercenter | #382 | D3 |
| hypocenter | C3 | #2844 |
| LCS | G ▷ C23×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ #382 | |
356 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 356 conjugacy classes of subgroups — too many to draw. Only its 133 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) = 384 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2,3)(6,10)(8,11)(12,14,15,18)(13,17,16,19);(5,9)(8,11)(14,18)(17,19)(20,22,23,27)(21,24,25,26);(2,3)(4,6)(5,8)(7,10)(9,11)(13,16)(14,18)(20,21,23,25)(22,26,27,24);(4,5,7,9)(6,8,10,11)(12,13,15,16)(14,17,18,19)Polycyclic presentation:
2^7,3|a^2=e,b^2=g,c^2=g,d^2=ef,b^a=bef,c^a=cf,c^b=cg,d^b=de,h^b=h^2,d^c=def54 conjugacy classes — view the full character table.
| maximal subgroup of | #10698257, #10698258, #10698260, … |
| maximal normal subgroup of | #10698257, #10698258, #10698260, … |
| maximal quotient of | #10725992, #10725994, #10726001, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 6794);