Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #85547 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C22×C42 ⋊ D3 |
| derived | C22⋊(C42⋊C3) ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22⋊(C42⋊C3) | C2 |
| Frattini | C22 | #1020 |
| Fitting | C22×C42 | D3 |
| socle | C24 | S4 |
| LCS | G ▷ C22⋊(C42⋊C3) | |
| UCS | C1 | |
160 conjugacy classes of subgroups; only the 10 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 10 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,4,3)(5,12,8,7)(6,13,10,14,11,9,16,15);(2,3,4)(5,6,9)(7,10,15)(8,11,13)(12,16,14)Polycyclic presentation:
2,3,2^6|e^2=g,f^2=gh,b^a=b^2,c^a=d,d^a=c,e^a=fh,f^a=eh,g^a=gh,c^b=d,d^b=cd,e^b=ef,f^b=egh,g^b=h,h^b=gh24 conjugacy classes — view the full character table.
| maximal subgroup of | #11711026, C2×#85547, #11715690, … |
| maximal normal subgroup of | #11711026, C2×#85547, #11715690, … |
| maximal quotient of | #11714236, #11714237, #11714238, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18123);