Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #226 |
| automorphism | Aut(G) = C22×C4×S4 (order 384) · Inn(G) = S4 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | #504 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C16 | S4 |
| commutator, hypocenter | SL(2,𝔽3) | C2×C8 |
| Frattini | C8 | C2×S4 |
| Fitting | #504 | D3 |
| socle | C2 | C8×S4 |
| prime radical | #985 | C4 |
| cosocle | #131⋊C3 | C22 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C16 | |
87 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,33,41,8,44,56,13,63,21,12,14,50,19,28,58,48,22,34,3,51,45,9,59,64,7,10,39,27,17,54,37,46,23,2,40,52,4,55,60,24,5,32,16,25,43,30,35,62)(6,11,15,26,18,29,36,47,20,31,38,49,42,53,57,61);(5,20)(6,22)(10,31)(11,33)(14,38)(15,40)(17,42)(18,44)(21,23)(25,49)(26,51)(28,53)(29,55)(32,34)(35,57)(36,59)(39,41)(43,45)(46,61)(47,63)(50,52)(54,56)(58,60)(62,64)Polycyclic presentation:
2^4,3,2^3|b^2=c,c^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 | #11711660, #11711661, C2×#73038, … |
| maximal normal subgroup of | #11711660, #11711661, C2×#73038, … |
| maximal quotient of | C16×GL(2,𝔽3), C16×#226, #11711383, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5614);