Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | #576 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #5415 |
| commutator, cosocle | #131⋊C3 | C22 |
| Frattini | C8 | C2×S4 |
| Fitting | #576 | D3 |
| hypercenter | Q16 | S4 |
| hypocenter | SL(2,𝔽3) | D8 |
| LCS | G ▷ #131⋊C3 ▷ #41⋊C3 ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ Q16 | |
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:
(1,35,23,12,24,37,7,10,25,2,5,34)(3,57,46,30,47,64,18,41,49,14,16,63)(4,31,53,9,43,59,15,51,54,19,20,60)(6,11,22,38)(8,39,55,13,36,61,29,48,56,26,27,62)(17,42,50,58)(21,52,44,32)(28,45,33,40);(2,8)(3,13)(4,15)(5,22)(6,24)(9,30)(10,33)(11,36)(12,29)(14,19)(16,45)(17,48)(18,26)(20,21)(23,25)(27,38)(28,35)(31,42)(32,41)(34,56)(37,55)(39,50)(40,47)(43,44)(46,62)(49,61)(51,58)(52,57)(59,63)(60,64)Polycyclic presentation:
2^4,3,2^3|b^2=h,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×#73137, #11714153, #11714154, … |
| maximal normal subgroup of | C2×#73137, #11714153, #11714154, … |
| maximal quotient of | #11711453, #11711458, #11711464, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5713);