Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #4621 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C2 |
| special linear | SL(2,ℤ/8) |
| symplectic | Sp(2,ℤ/8) |
| derived | #95⋊C3 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #857 |
| commutator, hypocenter | #95⋊C3 | C4 |
| Frattini, socle | C24 | S4 |
| Fitting | C2×#95 | D3 |
| prime radical, cosocle | C2×#95⋊C3 | C2 |
| LCS | G ▷ #95⋊C3 | |
| UCS | C1 ◁ C22 | |
106 conjugacy classes of subgroups; only the 15 normal subgroups are shown.
This group has 106 conjugacy classes of subgroups — too many to draw. Only its 15 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:
(1,7,6,3,4,2,8,5)(12,19,18,16,15,17,14,20);(2,3,8)(5,6,7)(9,12,17)(10,14,16)(11,15,19)(13,18,20)Polycyclic presentation:
2^2,3,2^5|a^2=b,d^2=f,e^2=fg,c^a=c^2,d^a=ef,e^a=dfg,f^a=fg,d^c=efh,e^c=deh,f^c=fg,g^c=fh,e^d=eh30 conjugacy classes — view the full character table.
| maximal subgroup of | C2×SL(2,ℤ/8), #11710978, #11710979, … |
| maximal normal subgroup of | C2×SL(2,ℤ/8), #11710978, #11710979, … |
| maximal quotient of | #11710943, #11710957, #11710958, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 572);