Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22 |
| direct product | C2×#72227559 |
| Fitting, Sylow | C2×#346283 ⋊ C3 |
| derived | #346283 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #72227559 |
| commutator, cosocle, hypocenter | #346283 | C2×C3 |
| Frattini | C25 | C2×C22⋊A4 |
| Fitting | C2×#346283 | C3 |
| socle | C23 | #73293 |
| LCS | G ▷ #346283 | |
| UCS | C1 ◁ C2 | |
2136 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 2136 conjugacy classes of subgroups — too many to draw. Only its 86 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:
(1,7,11,2,3,23)(4,13,35,8,15,32)(5,18,27)(6,20,40)(9,26,42)(10,14,29)(12,30,25)(16,24,28)(17,34,48)(19,37,45,22,38,43)(21,33,47)(31,44,36,41,46,39);(1,6,2,10)(3,7)(4,21,8,17)(5,19,9,22)(11,40,23,29)(12,24)(14,20)(16,36,25,39)(18,26)(27,45,42,43)(28,46,30,44)(31,41)(32,47,35,48)(37,38);(1,25,2,16)(3,26,7,18)(4,9,8,5)(6,36)(10,39)(11,32)(12,13,24,15)(14,38)(17,19)(20,37)(21,22)(23,35)(27,30)(28,42)(29,48,40,47)(31,34)(33,41)(43,46,45,44)64 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,7,11,2,3,23)(4,13,35,8,15,32)(5,18,27)(6,20,40)(9,26,42)(10,14,29)(12,30,25)(16,24,28)(17,34,48)(19,37,45,22,38,43)(21,33,47)(31,44,36,41,46,39), (1,6,2,10)(3,7)(4,21,8,17)(5,19,9,22)(11,40,23,29)(12,24)(14,20)(16,36,25,39)(18,26)(27,45,42,43)(28,46,30,44)(31,41)(32,47,35,48)(37,38), (1,25,2,16)(3,26,7,18)(4,9,8,5)(6,36)(10,39)(11,32)(12,13,24,15)(14,38)(17,19)(20,37)(21,22)(23,35)(27,30)(28,42)(29,48,40,47)(31,34)(33,41)(43,46,45,44));