Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#446 ⋊ C2 |
| Sylow | C3 ⋊ #3403 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D12 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#446 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #446 | D3 |
| hypocenter | C3 | #3403 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 ◁ #446 | |
380 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 121 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)(4,7,8,6)(5,10,9,11)(14,22)(17,21)(23,27)(25,26);(2,3)(4,6,8,7)(5,10,9,11)(13,18)(14,22)(16,24)(17,25)(19,20)(21,26);(4,5,8,9)(6,10,7,11)(12,14,15,22)(13,17,19,21)(16,23,24,27)(18,26,20,25);(2,3)(5,9)(10,11)(12,13)(14,21)(15,19)(16,18)(17,22)(20,24)(23,25)(26,27)Polycyclic presentation:
2^7,3|b^2=e,c^2=eg,d^2=eg,f^2=g,b^a=be,c^a=cf,f^a=fg,h^a=h^2,c^b=cg,d^b=de,f^c=fg,h^c=h^257 conjugacy classes — view the full character table.
| maximal subgroup of | #10708749, #10708753, #10708763, … |
| maximal normal subgroup of | #10708749, #10708753, #10708763, … |
| maximal quotient of | #10812075, #10814073, #10815009, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12532);