Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | Aut(G) = C22×C4×S4 (order 384) · Inn(G) = C2×S4 · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | #504 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C8 | C2×S4 |
| commutator, hypocenter | SL(2,𝔽3) | C2×C8 |
| Fitting | #504 | D3 |
| socle | C2 | C8×S4 |
| prime radical | #985 | C4 |
| cosocle | #131⋊C3 | C22 |
| hypercenter | C16 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C8 ◁ C16 | |
85 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,26,41,9,44,56,13,61,21,8,14,53,19,31,58,46,22,34,3,47,45,10,59,64,7,11,39,24,17,55,37,51,23,2,40,54,4,49,60,28,5,33,16,29,43,25,35,63)(6,12,15,30,18,32,36,52,20,27,38,48,42,50,57,62);(2,8)(5,20)(6,22)(9,24)(10,25)(11,12)(14,38)(15,40)(17,42)(18,44)(21,23)(26,27)(28,46)(29,30)(31,32)(35,57)(36,59)(39,41)(43,45)(47,48)(49,50)(51,52)(58,60)(61,62)Polycyclic presentation:
2^4,3,2^3|b^2=c,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 | C2×#73039, #11714137, #11714138, … |
| maximal normal subgroup of | C2×#73039, #11714137, #11714138, … |
| maximal quotient of | #11711379, #11711381, GL(2,𝔽3)⋊C16, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5615);