Finite Group DatabasebetaFinite Group Databasebeta| order | 12928 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C101 · Schur multiplier (H2) = C2×C4 |
| direct product | C101×#1391 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C101 | #99 |
| commutator | C23 | C42×C101 |
| Frattini, cosocle | C2×#41 | C22×C101 |
| socle | C2×C101 | #342 |
| prime radical | C2×#41×C101 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C101 ◁ C2×C4×C101 ◁ C2×#41×C101 ◁ G | |
166 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 166 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 133 |
| transitive degree | μt(G) = 3232 |
A minimal faithful permutation representation — 2 generators on 133 points:
(1,2,4,9)(3,11,12,23)(5,10,16,24)(6,8,15,21)(7,18,19,14)(13,25,26,32)(17,22,28,30)(20,29,31,27)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133);(1,3,5,13,6,14,17,27)(2,7,10,20,8,11,22,25)(4,12,16,26,15,18,28,29)(9,23,24,32,21,19,30,31)2626 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C101×#7398, C101×#7399, C101×#7406, … |
| maximal normal subgroup of | C101×#7398, C101×#7399, C101×#7406, … |
| maximal quotient of | C101×#7188, C101×#7204, C101×#7214, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12928, 123);