Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #226 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C8 | S4 |
| commutator, hypocenter | SL(2,𝔽3) | C42 |
| Frattini | C2×C4 | C2×S4 |
| Fitting | C2×#131 | D3 |
| socle | C22 | C4×S4 |
| prime radical, cosocle | C2×#41⋊C3 | C22 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C8 | |
149 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 149 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 36 points:
(5,27,32,11,8,26,17,22,21,6,31,36,10,12,30,24,20,28,7,35,19,14,15,34)(9,13,16,23,18,25,29,33);(1,2,3,4)(8,18)(9,20)(12,25)(13,27)(15,29)(16,31)(19,21)(22,33)(23,35)(26,28)(30,32)(34,36)Polycyclic presentation:
2^4,3,2^3|a^2=c,b^2=d,d^2=h,f^2=h,g^2=h,e^a=e^2,f^a=gh,g^a=fh,f^e=g,g^e=fgh,g^f=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | #11711488, #11711489, #11711544, … |
| maximal normal subgroup of | #11711488, #11711489, #11711544, … |
| maximal quotient of | #11710990, #11710992, C8×GU(2,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5697);