Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C22×D8 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #5415 |
| commutator, cosocle | C8×A4 | C22 |
| Frattini | C8 | C2×S4 |
| Fitting | C22×D8 | D3 |
| socle | C23 | #213 |
| hypercenter | D8 | S4 |
| hypocenter | A4 | D16 |
| LCS | G ▷ C8×A4 ▷ C4×A4 ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ D8 | |
112 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 18 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,4)(6,10)(7,14)(8,15)(11,12)(13,19)(16,20)(17,18);(2,3)(5,6)(7,10)(8,17)(9,13)(11,14)(12,15)(16,19)(18,20)Polycyclic presentation:
2^5,3,2^2|c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^2,g^a=h,h^a=g,c^b=cd,d^b=de,g^f=h,h^f=gh29 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#73133, #11714144, #11714145, … |
| maximal normal subgroup of | C2×#73133, #11714144, #11714145, … |
| maximal quotient of | #11711453, #11711457, #11711462, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5709);