Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | #578 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D4×S4 |
| commutator, cosocle | #41⋊C3 | C23 |
| Frattini | C4 | C22×S4 |
| Fitting | #578 | D3 |
| hypercenter | D8 | S4 |
| hypocenter | SL(2,𝔽3) | C2×D4 |
| LCS | G ▷ #41⋊C3 ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D8 | |
245 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 245 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,24,26)(2,11,36,9,25,35)(3,41,45,14,44,62)(4,20,52,19,47,50)(5,23,7)(8,57,56,28,37,64)(10,51,60)(12,22)(13,53,55,32,29,63)(15,16,61,18,39,43)(17,40)(21,48)(30,54)(31,33,59)(38,58)(42,46);(1,3)(2,8)(4,14)(5,16)(6,17)(7,18)(9,13)(10,32)(11,29)(12,30)(15,19)(20,39)(21,40)(22,38)(23,43)(24,44)(25,37)(26,45)(27,46)(28,31)(33,57)(34,58)(35,55)(36,56)(41,47)(42,48)(49,54)(50,61)(51,53)(52,62)(59,64)(60,63);(1,2)(3,13)(4,10)(5,22)(6,25)(7,9)(8,18)(11,27)(12,24)(14,28)(15,32)(16,30)(17,29)(19,31)(20,49)(21,51)(23,36)(26,35)(33,48)(34,47)(37,46)(38,44)(39,58)(40,57)(41,54)(42,53)(43,55)(45,56)(50,60)(52,59)(61,64)(62,63)Polycyclic presentation:
2^4,3,2^3|d^2=h,f^2=h,g^2=h,b^a=bh,c^a=ch,e^a=e^2,f^a=gh,g^a=fh,c^b=cd,d^b=dh,d^c=dh,f^e=g,g^e=fgh,g^f=gh31 conjugacy classes — view the full character table.
| maximal subgroup of | #11714212, #11714213, #11714214, … |
| maximal normal subgroup of | #11714212, #11714213, #11714214, … |
| maximal quotient of | #11712366, #11712372, #11712381, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18051);