Finite Group DatabasebetaFinite Group Databasebeta| order | 12928 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C101 · Schur multiplier (H2) = C23 |
| direct product | C4×D16×C101 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C101 | D8 |
| commutator | C8 | C22×C4×C101 |
| Frattini, cosocle | C2×C8 | C23×C101 |
| socle | C22×C101 | C2×D8 |
| prime radical | C2×D16×C101 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C101 ◁ C42×C101 ◁ C4×C8×C101 ◁ G | |
174 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 174 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 6464 |
A minimal faithful permutation representation — 3 generators on 121 points:
(1,2,3,4)(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);(5,6)(7,10)(8,17)(9,13)(11,14)(12,15)(16,19)(18,20);(6,10)(7,14)(8,15)(11,12)(13,19)(16,20)(17,18)4444 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C101×#7383, C101×#7385, C8×D16×C101, … |
| maximal normal subgroup of | C101×#7383, C101×#7385, C8×D16×C101, … |
| maximal quotient of | C8×D16×C101, C101×#12069, C101×#12081, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12928, 904);