Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | #576 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D4×S4 |
| commutator, cosocle | #41⋊C3 | C23 |
| Frattini | C4 | C22×S4 |
| Fitting | #576 | D3 |
| hypercenter | Q16 | S4 |
| hypocenter | SL(2,𝔽3) | C2×D4 |
| LCS | G ▷ #41⋊C3 ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q16 | |
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:
(1,43,24,17,25,44,7,15,26,3,5,42)(2,55,34,29,35,56,12,27,36,8,10,54)(4,39,50,14,51,63,22,47,52,19,20,64)(6,16,23,41)(9,37,59,13,45,61,18,57,60,30,31,62)(11,28,33,53)(21,48,49,40)(32,58,46,38);(1,2,7,12)(3,13,17,30)(4,18,22,9)(5,10,25,35)(6,11,23,33)(8,19,29,14)(15,37,43,57)(16,38,41,58)(20,45,51,31)(21,46,49,32)(24,34,26,36)(27,47,55,39)(28,48,53,40)(42,61,44,62)(50,60,52,59)(54,64,56,63);(5,23)(6,25)(10,33)(11,35)(15,41)(16,43)(20,49)(21,51)(24,26)(27,53)(28,55)(31,46)(32,45)(34,36)(37,58)(38,57)(39,48)(40,47)(42,44)(50,52)(54,56)(59,60)(61,62)(63,64)Polycyclic presentation:
2^4,3,2^3|b^2=h,c^2=h,d^2=h,f^2=h,g^2=h,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 | #11714211, #11714213, #11714215, … |
| maximal normal subgroup of | #11714211, #11714213, #11714215, … |
| maximal quotient of | #11712369, #11712375, #11712382, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18042);