Finite Group DatabasebetaFinite Group Databasebeta| order | 12928 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C101 · Schur multiplier (H2) = C22×C4 |
| direct product | C101×#1975 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C101 | C2×D4 |
| commutator | C2×C4 | C22×C4×C101 |
| Frattini, cosocle | C22×C4 | C23×C101 |
| socle | C22×C101 | C4×D4 |
| prime radical | C2×#135×C101 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C101 ◁ C22×C8×C101 ◁ G | |
292 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 292 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) = 121 |
| transitive degree | μt(G) = 3232 |
A minimal faithful permutation representation — 3 generators on 121 points:
(1,2,3,4)(5,6,9,13,10,14,8,11)(7,15,18,12,16,19,17,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121);(2,4)(6,11,14,13)(7,16)(12,19,20,15)(17,18);(5,7,9,18,10,16,8,17)(6,12,11,15,14,20,13,19)3232 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C101×#7109, C101×#13024, C101×#13037, … |
| maximal normal subgroup of | C101×#7109, C101×#13024, C101×#13037, … |
| maximal quotient of | C101×#9658, C101×#9659, C101×#9660, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12928, 707);