Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting, Sylow | H(7,𝔽2) ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2 | C24×A4 |
| commutator, cosocle, hypocenter | Q8 | C24×C3 |
| Fitting | H(7,𝔽2) | C3 |
| hypercenter | 2−1+4 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C2 ◁ 2−1+4 | |
1232 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,5,19)(2,27,28,11,12,42)(3,15,31)(4,18,34)(6,22,35)(7,20,37)(8,21,36)(9,38,39,23,24,47)(10,33,41,17,26,46)(13,40,43,25,29,48)(14,30,44)(16,32,45);(1,6)(2,25)(3,4)(5,32)(7,16)(8,14)(9,10)(11,13)(12,24)(15,21)(17,23)(18,22)(19,36)(20,30)(26,29)(27,38)(33,40)(34,37)(39,47)(43,48);(1,3,8,16)(2,9,11,23)(4,6,7,14)(5,15,21,32)(10,25,17,13)(12,24,27,38)(18,22,20,30)(19,31,36,45)(26,40,33,29)(28,39,42,47)(34,35,37,44)(41,48,46,43);(1,2,8,11)(3,9,16,23)(4,17,7,10)(5,12,21,27)(6,13,14,25)(15,24,32,38)(18,33,20,26)(19,28,36,42)(22,29,30,40)(31,39,45,47)(34,46,37,41)(35,43,44,48)Polycyclic presentation:
2^4,3,2^3|a^2=h,b^2=h,c^2=h,f^2=h,g^2=h,c^a=ch,d^a=dh,c^b=ch,f^e=fg,g^e=f,g^f=gh67 conjugacy classes — view the full character table.
| maximal subgroup of | #60388⋊C3, #60389⋊C3, #11710897, … |
| maximal normal subgroup of | #60388⋊C3, #60389⋊C3, #11710897, … |
| maximal quotient of | #11710601, #11710605, #62851⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20112);