Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C2×H(5,𝔽2) ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×GL(2,ℤ/4) |
| commutator, cosocle | #41⋊C3 | C23 |
| Frattini | C4 | C22×S4 |
| Fitting | C2×H(5,𝔽2) | D3 |
| hypercenter | C2×Q8 | S4 |
| hypocenter | SL(2,𝔽3) | C2×D4 |
| LCS | G ▷ #41⋊C3 ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×Q8 | |
300 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 300 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) = 48 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,15,48,19,33,44,7,24,36,4,22,21)(2,11,39,26,46,35,13,29,31,9,27,40)(3,5,43,16,37,32,6,23,47,14,18,42)(8,28,20,12,34,45,10,25,41,17,30,38);(1,3,7,6)(2,8,13,10)(4,16,19,14)(5,15,23,24)(9,12,26,17)(11,25,29,28)(18,33,37,22)(20,31,41,39)(21,32,44,42)(27,30,46,34)(35,38,40,45)(36,47,48,43);(1,2)(3,8)(4,17)(5,20)(6,10)(7,13)(9,16)(11,21)(12,19)(14,26)(15,31)(18,35)(22,45)(23,41)(24,39)(25,32)(27,47)(28,42)(29,44)(30,48)(33,38)(34,36)(37,40)(43,46)Polycyclic presentation:
2^4,3,2^3|b^2=h,c^2=h,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,c^b=ch,d^c=dh,f^e=g,g^e=fgh,g^f=gh31 conjugacy classes — view the full character table.
| maximal subgroup of | #11713172, #11713173, #11713196, … |
| maximal normal subgroup of | #11713172, #11713173, #11713196, … |
| maximal quotient of | #11712068, #11712069, #11712070, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18102);