Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #226 |
| automorphism | Aut(G) = C2×C8×S4 (order 384) · Inn(G) = S4 · Out(G) = C2×C8 |
| homology | Abelianization (H1) = C16 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C16 | S4 |
| commutator, hypocenter | SL(2,𝔽3) | C16 |
| Fitting | #504 | D3 |
| socle | C2 | A4⋊C16 |
| prime radical | #41⋊C3 | C8 |
| cosocle | #504⋊C3 | C2 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C16 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 128 points:
(1,30,57,36,23,39,101,98,5,72,103,80,62,84,126,123,8,12,59,56,4,66,99,77,26,42,105,102,20,109,124,113)(2,47,82,22,41,50,116,74,11,93,117,61,86,95,128,7,14,17,83,68,10,89,115,25,44,53,118,111,38,121,127,32)(3,58,64,40,52,76,73,69,16,104,107,85,97,112,28,13,19,35,67,63,15,100,70,43,55,79,110,106,49,125,31,27)(6,48,45,9,65,88,51,78,24,94,91,34,108,120,96,114,29,46,18,37,21,90,87,33,71,92,54,81,60,122,119,75);(6,7,31)(12,13,48)(17,18,59)(21,22,67)(24,25,73)(27,46,30)(28,29,32)(35,36,83)(39,40,90)(42,43,94)(45,57,47)(50,51,101)(53,54,105)(56,82,58)(60,61,110)(63,88,66)(64,65,68)(69,92,72)(70,71,74)(76,77,116)(79,80,118)(84,85,122)(87,99,89)(91,103,93)(95,96,126)(98,115,100)(102,117,104)(106,120,109)(107,108,111)(112,113,128)(119,124,121)(123,127,125)Polycyclic presentation:
2^4,3,2^3|a^2=b,b^2=c,c^2=d,d^2=h,f^2=h,g^2=h,e^a=e^2,f^a=gh,g^a=fh,f^e=g,g^e=fgh,g^f=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#68006, #11711657, #11711660, … |
| maximal normal subgroup of | C2×#68006, #11711657, #11711660, … |
| maximal quotient of | SL(2,𝔽3)⋊C32, C2×#68006, #11919934, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 582);