Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | #578 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #5415 |
| commutator, cosocle | #131⋊C3 | C22 |
| Frattini | C8 | C2×S4 |
| Fitting | #578 | D3 |
| hypercenter | D8 | S4 |
| hypocenter | SL(2,𝔽3) | D8 |
| LCS | G ▷ #131⋊C3 ▷ #41⋊C3 ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ D8 | |
95 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,24,26)(2,46,37,8,25,56)(3,59,47,14,45,64)(4,20,52,15,41,51)(5,23,7)(9,49,60,10,54,62)(11,55,13,29,36,31)(12,30)(16,63,18,38,44,40)(17,39)(19,32,61,28,34,58)(21,42)(22,43)(33,35)(48,57)(50,53);(1,2)(3,8)(4,19)(5,22)(6,25)(7,13)(9,14)(10,15)(11,27)(12,24)(16,43)(17,46)(18,31)(20,35)(21,34)(23,37)(26,36)(28,40)(29,48)(30,45)(32,57)(33,59)(38,53)(39,54)(41,50)(42,49)(44,56)(47,55)(51,61)(52,62)(58,64)(60,63)Polycyclic presentation:
2^4,3,2^3|c^2=dh,d^2=h,f^2=h,g^2=h,b^a=bc,c^a=cd,d^a=dh,e^a=e^2,f^a=gh,g^a=fh,c^b=cd,d^b=dh,f^e=g,g^e=fgh,g^f=gh25 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#73139, #11714153, #11714155, … |
| maximal normal subgroup of | C2×#73139, #11714153, #11714155, … |
| maximal quotient of | #11711454, #11711457, #11711465, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5715);