Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#442 ⋊ C2 |
| Sylow | C3 ⋊ #3176 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#442 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #442 | D3 |
| hypocenter | C3 | #3176 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #442 | |
386 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 386 conjugacy classes of subgroups — too many to draw. Only its 115 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)(4,6,7,10)(5,9,8,11)(13,18)(17,25)(19,24)(23,27);(2,3)(12,14,21,26,16,22,15,20)(13,17,24,27,18,25,19,23);(2,3)(5,8)(9,11)(12,13,16,18)(14,17,22,25)(15,19,21,24)(20,23,26,27);(4,5)(6,9)(7,8)(10,11)(14,20)(15,21)(17,23)(19,24)(22,26)(25,27)Polycyclic presentation:
2^7,3|b^2=g,c^2=fg,d^2=e,f^2=g,b^a=be,c^a=cf,f^a=fg,d^b=dg,h^b=h^2,h^c=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10950455, D3×#3176, C2×#79860, … |
| maximal normal subgroup of | #10950455, D3×#3176, C2×#79860, … |
| maximal quotient of | #10795757, #10799467, #10799548, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12436);