Finite Group DatabasebetaFinite Group Databasebeta| order | 12928 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C101 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#491×C101 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C101 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C101 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C101 ◁ C2×C42×C101 ◁ G | |
384 conjugacy classes of subgroups; only the 200 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 200 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 12928 |
A minimal faithful permutation representation — 4 generators on 127 points:
(1,2)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127);(3,4)(5,7)(6,8)(9,10)(11,12,17,22)(13,19,24,25)(14,18,23,26)(15,21,16,20);(3,5,6,9)(4,7,8,10)(11,13,15,14,17,24,16,23)(12,18,20,19,22,26,21,25);(4,8)(7,10)(11,14,15,24,17,23,16,13)(12,19,20,26,22,25,21,18)3232 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C101×#9004, C101×#10083, C101×#10084, … |
| maximal normal subgroup of | C101×#9004, C101×#10083, C101×#10084, … |
| maximal quotient of | C2×C101×#1836, C2×C101×#1935, C2×C101×#1960, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12928, 1866);