Finite Group DatabasebetaFinite Group Databasebeta| order | 12928 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C101 · Schur multiplier (H2) = C23 |
| direct product | C2×#349×C101 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C101 | C22⋊C4 |
| commutator | C22 | C22×C8×C101 |
| Frattini, cosocle | C2×C8 | C23×C101 |
| socle | C22×C101 | C22⋊C8 |
| prime radical | C24×C101 | C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C101 ◁ C22×C8×C101 ◁ G | |
236 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 236 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 3232 |
A minimal faithful permutation representation — 3 generators on 119 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101)(102,103);(104,105,106,110,107,111,113,117,108,109,112,115,114,116,118,119);(105,109)(106,112)(111,116)(113,118)4444 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C101×#7191, C101×#7192, C101×#7193, … |
| maximal normal subgroup of | C101×#7191, C101×#7192, C101×#7193, … |
| maximal quotient of | C2×C101×#1314, C101×#11418, C2×C101×#1322, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12928, 846);