Finite Group DatabasebetaFinite Group Databasebeta| order | 2304 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C4 · Schur multiplier (H2) = C22×C3 |
| direct product | C4×(#41⋊C3)⋊A4 |
| Fitting, Sylow | C4×#585 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C42 | A42 |
| commutator, hypocenter | H(5,𝔽2) | C2×C32×C4 |
| Frattini, socle | C22 | C22×A42 |
| Fitting | C4×#585 | C32 |
| prime radical | C2×(#41⋊C3)⋊A4 | C2 |
| cosocle | C2×H(5,𝔽2) | C22×C32 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C42 | |
1170 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 1170 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 2 generators on 52 points:
(5,35,9,6,22,13,10,33,25,14,24,36)(7,48,17,11,21,28,20,32,18,31,41,29)(8,34,23,12)(15,50,38,26,39,45,40,46,51,47,43,52)(16,30,19,27)(37,49,42,44);(1,2,3,4)(5,19,40,9,7,43,10,16,15,25,20,39)(6,30,47,13,11,50,14,27,26,36,31,46)(8,17,51,24,21,37,23,18,38,22,41,42)(12,28,52,35,32,44,34,29,45,33,48,49)200 conjugacy classes — view the full character table.
| maximal subgroup of | #72228568, #72253041, C4×#11923520, … |
| maximal normal subgroup of | #72228568, #72253041, C4×#11923520, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,35,9,6,22,13,10,33,25,14,24,36)(7,48,17,11,21,28,20,32,18,31,41,29)(8,34,23,12)(15,50,38,26,39,45,40,46,51,47,43,52)(16,30,19,27)(37,49,42,44), (1,2,3,4)(5,19,40,9,7,43,10,16,15,25,20,39)(6,30,47,13,11,50,14,27,26,36,31,46)(8,17,51,24,21,37,23,18,38,22,41,42)(12,28,52,35,32,44,34,29,45,33,48,49));