Finite Group DatabasebetaFinite Group Databasebeta| order | 12928 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C101 · Schur multiplier (H2) = C25 |
| direct product | C101×#1889 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C101 | #120 |
| commutator | C23 | C22×C4×C101 |
| Frattini, cosocle | C24 | C23×C101 |
| prime radical | C2×#120×C101 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C101 ◁ C24×C101 ◁ G | |
578 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 578 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 1616 |
A minimal faithful permutation representation — 3 generators on 113 points:
(1,2,4,3)(5,6)(7,11,12,9)(8,10)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113);(1,3,4,2)(5,6)(7,9)(8,10)(11,12);(5,7)(6,9)(8,12)(10,11)3232 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C101×#12753, C101×#12759, C101×#12761, … |
| maximal normal subgroup of | C101×#12753, C101×#12759, C101×#12761, … |
| maximal quotient of | C101×#8965, C101×#8967, C101×#9009, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12928, 621);