Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C2×#131 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C8 | C2×S4 |
| commutator, hypocenter | SL(2,𝔽3) | C22×C4 |
| Frattini | C4 | C22×S4 |
| Fitting | C2×#131 | D3 |
| socle | C2 | C2×C4×S4 |
| prime radical | #5923 | C2 |
| cosocle | #41⋊C3 | C23 |
| hypercenter | C2×C8 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C8 ◁ C2×C8 | |
248 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,40,48,13,5,39,19,35,24,3,46,61,7,14,45,37,22,42,4,59,21,16,17,58)(2,41,56,28,11,53,27,51,34,9,47,64,8,29,55,50,23,54,10,60,33,26,31,63)(6,15,18,36,25,43,49,62)(12,30,32,52,20,38,44,57);(2,8)(9,26)(10,27)(11,23)(12,20)(28,50)(29,41)(30,38)(31,47)(32,44)(33,34)(51,60)(52,57)(53,54)(55,56)(63,64);(1,2)(3,9)(4,10)(5,20)(6,23)(7,8)(11,25)(12,22)(13,28)(14,38)(15,41)(16,26)(17,44)(18,47)(19,27)(21,34)(24,33)(29,43)(30,40)(31,49)(32,46)(35,57)(36,60)(37,50)(39,54)(42,53)(45,56)(48,55)(51,62)(52,59)(58,64)(61,63)Polycyclic presentation:
2^4,3,2^3|c^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 | #11711560, #11711565, #11712836, … |
| maximal normal subgroup of | #11711560, #11711565, #11712836, … |
| maximal quotient of | C8×#986, C8×#983, C8×#984, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18024);