Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C23×D4 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×S4 |
| commutator, cosocle | C2×A4 | C24 |
| Fitting | C23×D4 | D3 |
| socle | C23 | C23×D3 |
| hypercenter | C2×D4 | S4 |
| hypocenter | A4 | H(5,𝔽2) |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2 ◁ C2×D4 | |
907 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 907 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 4 generators on 12 points:
(1,2,4)(6,10)(9,12);(7,11)(9,12);(5,7)(6,9)(8,11)(10,12);(2,3)(5,6)(7,9)(8,10)(11,12)Polycyclic presentation:
2^5,3,2^2|d^a=de,f^a=f^2,g^a=h,h^a=g,c^b=ce,g^f=h,h^f=gh44 conjugacy classes — view the full character table.
| maximal subgroup of | #11713202, #11713203, #11713205, … |
| maximal normal subgroup of | #11713202, #11713203, #11713205, … |
| maximal quotient of | #11714311, #11714314, #11714315, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20053);