Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C22×S4 |
| commutator, prime radical, cosocle | C2×SL(2,𝔽3) | C23 |
| Fitting | #557 | D3 |
| hypercenter | C4⋊C4 | S4 |
| hypocenter | SL(2,𝔽3) | #41 |
| LCS | G ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
184 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 184 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) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,4,5,2)(3,6,7,8)(9,22,27)(10,17,32,16,26,31)(11,20,38,19,33,36)(12,13,25)(15,37,40)(18,24)(21,34)(28,29,39);(1,3,5,7)(2,6,4,8)(9,11,12,19)(10,15,16,28)(13,20,22,33)(14,21,23,34)(17,29,26,37)(18,30,24,35)(25,36,27,38)(31,39,32,40);(1,2,5,4)(3,6,7,8)(9,10)(11,15)(12,16)(13,24)(14,26)(17,23)(18,22)(19,28)(20,35)(21,37)(25,32)(27,31)(29,34)(30,33)(36,40)(38,39)Polycyclic presentation:
2^4,3,2^3|a^2=dh,b^2=d,c^2=dh,f^2=h,g^2=h,c^a=ch,e^a=e^2,f^a=gh,g^a=fh,c^b=cd,f^e=g,g^e=fgh,g^f=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #11712294, #11712298, #11712349, … |
| maximal normal subgroup of | #11712294, #11712298, #11712349, … |
| maximal quotient of | #11712975, #11713039, #11713046, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17993);