Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×C8×D4 ⋊ C2 |
| Sylow | C3 ⋊ #2968 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C23×D3 |
| commutator, socle | C22×C3 | C23×C4 |
| Fitting | C3×C8×D4 | C2 |
| prime radical | #939 | C4 |
| cosocle | C2×C3×C4 | C24 |
| hypercenter | C8×D4 | D3 |
| hypocenter | C3 | #2968 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ C8×D4 | |
372 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 27 points:
(2,3)(4,6,7,10)(5,9,8,11)(14,20)(17,23)(21,26)(24,27);(1,2,3)(12,14,15,21,16,20,22,26)(13,17,18,24,19,23,25,27);(5,8)(9,11);(4,5)(6,9)(7,8)(10,11)(12,13,15,18,16,19,22,25)(14,17,21,24,20,23,26,27)Polycyclic presentation:
2^7,3|a^2=f,c^2=f,d^2=e,f^2=g,b^a=be,d^c=dg,h^d=h^290 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#2968, C2×#79402, #11659725, … |
| maximal normal subgroup of | D3×#2968, C2×#79402, #11659725, … |
| maximal quotient of | #10793448, #10793450, #10793479, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 11978);