Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#459 ⋊ C2 |
| Sylow | C3 ⋊ #3305 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#459 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #459 | D3 |
| hypocenter | C3 | #3305 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #459 | |
408 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 408 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)(6,11)(8,10)(12,14,16,17)(13,18,20,24)(15,21,23,25)(19,26,27,22);(2,3)(12,15,16,23)(13,19,20,27)(14,21,17,25)(18,26,24,22);(2,3)(4,6)(5,8)(7,11)(9,10)(12,14,16,17)(13,18,20,24)(15,21,23,25)(19,26,27,22);(2,3)(4,5,7,9)(6,10,11,8)(12,13,16,20)(14,18,17,24)(15,22,23,26)(19,25,27,21)Polycyclic presentation:
2^7,3|a^2=e,b^2=g,c^2=g,d^2=g,f^2=g,b^a=beg,c^a=cf,f^a=fg,h^a=h^2,d^b=deg,h^b=h^2,f^c=fg,h^c=h^2,h^d=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10784949, #10784957, #10784964, … |
| maximal normal subgroup of | #10784949, #10784957, #10784964, … |
| maximal quotient of | #10810659, #10811450, #10812356, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12929);