Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C3×#476 ⋊ C2 |
| Sylow | C3 ⋊ #3341 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D3×D4 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#476 | C2 |
| socle | C22×C3 | C22×D4 |
| hypercenter | #476 | D3 |
| hypocenter | C3 | #3341 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #476 | |
364 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 364 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)(9,10)(12,14,16,17)(13,18,20,24)(15,21,23,25)(19,26,27,22);(4,7,8,5)(6,10,11,9)(12,15)(13,19)(14,21)(16,23)(17,25)(18,26)(20,27)(22,24);(2,3)(4,6)(5,9)(7,10)(8,11)(12,14,16,17)(13,18,20,24)(15,21,23,25)(19,26,27,22);(4,5,8,7)(6,10,11,9)(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=eg,d^2=g,f^2=g,b^a=beg,c^a=cf,f^a=fg,d^b=deg,h^b=h^2,f^c=fg,h^d=h^254 conjugacy classes — view the full character table.
| maximal subgroup of | #10784831, #10784832, #10784838, … |
| maximal normal subgroup of | #10784831, #10784832, #10784838, … |
| maximal quotient of | #10811450, #10811456, #10812844, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 13658);