Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C2×#131 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C4⋊S4 |
| commutator, cosocle | #41⋊C3 | C23 |
| Frattini | C4 | C22×S4 |
| Fitting | C2×#131 | D3 |
| socle | C2 | C2×C4⋊S4 |
| hypercenter | C2×C8 | S4 |
| hypocenter | SL(2,𝔽3) | C2×D4 |
| LCS | G ▷ #41⋊C3 ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×C8 | |
246 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 246 conjugacy classes of subgroups — too many to draw. Only its 36 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:
(1,51,41,18,6,50,3,47,29,4,15,61,8,19,43,14,27,53,5,38,26,21,23,62)(2,56,58,17,11,59,22,44,37,10,42,64,13,33,57,46,28,60,9,52,36,35,31,63)(7,20,24,48,30,54,16,39)(12,34,32,45,25,55,40,49);(1,3,8,5)(2,9,13,22)(4,14,21,18)(6,15,27,23)(7,16,30,24)(10,17,35,46)(11,31,28,42)(12,32,25,40)(19,38,51,47)(20,39,54,48)(26,41,29,43)(33,44,56,52)(34,45,55,49)(36,57,37,58)(50,61,53,62)(59,63,60,64);(1,2)(3,9)(4,17)(5,22)(6,25)(7,28)(8,13)(10,18)(11,30)(12,27)(14,35)(15,40)(16,42)(19,49)(20,52)(21,46)(23,32)(24,31)(26,37)(29,36)(33,39)(34,38)(41,58)(43,57)(44,54)(45,51)(47,55)(48,56)(50,64)(53,63)(59,61)(60,62)Polycyclic presentation:
2^4,3,2^3|b^2=h,c^2=d,d^2=h,f^2=h,g^2=h,c^a=cd,d^a=dh,e^a=e^2,f^a=gh,g^a=fh,f^e=g,g^e=fgh,g^f=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #11711569, #11711575, #11711606, … |
| maximal normal subgroup of | #11711569, #11711575, #11711606, … |
| maximal quotient of | #11712161, #11712165, #11712170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18027);