Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C2×H(5,𝔽2) ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2 | C23×S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C24 |
| Fitting | C2×H(5,𝔽2) | D3 |
| hypercenter | C2×Q8 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C2×Q8 | |
734 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 734 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,31,33)(2,29,35,10,46,45)(3,17,42)(4,34,47)(5,36,16)(6,14,41)(7,20,48)(8,27,43,12,44,39)(9,30,37,22,25,18)(11,38,24,28,32,26)(13,21,40)(15,23,19);(1,4,7,6)(2,9,10,22)(3,13,15,16)(5,17,21,23)(8,24,12,26)(11,27,28,44)(14,31,34,20)(18,35,37,45)(19,36,42,40)(25,29,30,46)(32,39,38,43)(33,47,48,41);(1,3,7,15)(2,8,10,12)(4,16,6,13)(5,14,21,34)(9,26,22,24)(11,25,28,30)(17,20,23,31)(18,32,37,38)(19,33,42,48)(27,46,44,29)(35,43,45,39)(36,41,40,47);(1,2)(3,8)(4,9)(5,18)(6,22)(7,10)(11,19)(12,15)(13,24)(14,32)(16,26)(17,35)(20,43)(21,37)(23,45)(25,33)(27,36)(28,42)(29,47)(30,48)(31,39)(34,38)(40,44)(41,46)Polycyclic presentation:
2^4,3,2^3|b^2=h,c^2=h,f^2=h,g^2=h,d^a=dh,e^a=e^2,f^a=gh,g^a=fh,c^b=ch,f^e=g,g^e=fgh,g^f=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | #11713172, #11713174, #11713370, … |
| maximal normal subgroup of | #11713172, #11713174, #11713370, … |
| maximal quotient of | #11714607, #11714645, #11714652, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20069);