Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | #393 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #1020⋊C2 |
| commutator, cosocle, hypocenter | C22⋊SL(2,𝔽3) | C22 |
| Frattini, socle | C23 | C2×S4 |
| Fitting | #393 | D3 |
| LCS | G ▷ C22⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
126 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 126 conjugacy classes of subgroups — too many to draw. Only its 12 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,5,3,6,4,2)(7,8)(9,19,21)(10,15,24,14,20,23)(11,18,13)(16,17);(3,8)(5,6)(9,10)(11,17)(12,20)(13,14)(15,22)(16,19)(18,24)(21,23)Polycyclic presentation:
2^2,3,2^5|d^2=h,e^2=h,b^a=bh,c^a=c^2,d^a=eh,e^a=dh,f^a=fg,d^b=dfg,e^b=ef,d^c=e,e^c=deh,f^c=g,g^c=fg,e^d=eh21 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#73000, #11711932, #11711937, … |
| maximal normal subgroup of | C2×#73000, #11711932, #11711937, … |
| maximal quotient of | #11711131, #11711141, #11711143, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5576);