Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C2×S4 |
| commutator, hypocenter | SL(2,𝔽3) | C42 |
| Fitting | C2×#131 | D3 |
| socle | C22 | C4×S4 |
| prime radical, cosocle | C2×#41⋊C3 | C22 |
| hypercenter | C2×C8 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C4 ◁ C2×C8 | |
143 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 143 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:
(5,23,32,12,8,27,17,25,21,6,31,36,10,13,30,22,20,28,7,33,19,11,15,35)(9,14,16,26,18,24,29,34);(1,2,3,4)(6,11)(8,18)(9,20)(12,22)(13,14)(15,29)(16,31)(19,21)(23,24)(25,26)(30,32)(33,34)Polycyclic presentation:
2^4,3,2^3|a^2=c,b^2=d,d^2=h,f^2=h,g^2=h,b^a=bh,e^a=e^2,f^a=gh,g^a=fh,f^e=g,g^e=fgh,g^f=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #11711491, C2×#73123, #11712565, … |
| maximal normal subgroup of | #11711491, C2×#73123, #11712565, … |
| maximal quotient of | #11710991, #11710992, #11710995, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5699);