Finite Group DatabasebetaFinite Group Databasebeta| order | 2368 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C37 · Schur multiplier (H2) = C22 |
| direct product | C37×#471 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C37 | #121 |
| commutator, Frattini, cosocle | C2×C4 | C23×C37 |
| prime radical | AGL(1,ℤ/8)×C37 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C37 ◁ C2×C4×C37 ◁ G | |
104 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 104 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 592 |
A minimal faithful permutation representation — 3 generators on 53 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37)(39,43)(40,46)(41,49)(45,51)(47,48)(50,52);(38,39,41,43,42,45,49,51)(40,44,47,50,48,53,46,52);(38,40)(39,44)(41,47)(42,48)(43,50)(45,53)(46,49)(51,52)592 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C37×#471, C37×#3067, C37×#3068, … |
| maximal normal subgroup of | C2×C37×#471, C37×#3067, C37×#3068, … |
| maximal quotient of | C37×#1665, C37×#1666, C37×#1671, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2368, 152);