Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C8 | C2×S4 |
| commutator, hypocenter | SL(2,𝔽3) | C2×C8 |
| Fitting | C2×#131 | D3 |
| socle | C2 | C2×A4⋊C8 |
| prime radical | C2×#41⋊C3 | C4 |
| cosocle | #131⋊C3 | C22 |
| hypercenter | C2×C8 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C8 ◁ C2×C8 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,22,24)(2,11,36,8,23,34)(3,35,43)(4,46,48)(5,21,7)(9,29,54,26,42,53)(10,31,58,27,47,56)(12,20)(13,57,62)(14,41,16)(17,45,19)(28,51,64,50,61,63)(30,40)(32,44)(37,60,39)(52,59);(1,2,3,9,4,10,13,28,7,8,16,26,19,27,39,50)(5,20,14,40,17,44,37,59,22,12,35,30,46,32,57,52)(6,23,15,42,18,47,38,61,25,11,33,29,49,31,55,51)(21,36,41,54,45,58,60,64,24,34,43,53,48,56,62,63)Polycyclic presentation:
2^4,3,2^3|a^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 | #11711485, #11711491, #11711498, … |
| maximal normal subgroup of | #11711485, #11711491, #11711498, … |
| maximal quotient of | #11711348, #11711350, #11711351, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5695);