Finite Group DatabasebetaFinite Group Databasebeta| order | 12928 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C101 · Schur multiplier (H2) = C26 |
| direct product | C2×#458×C101 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C101 | #120 |
| commutator, Frattini, cosocle | C23 | C24×C101 |
| prime radical | C2×H(5,𝔽2)×C101 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C101 ◁ C24×C101 ◁ G | |
794 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 794 conjugacy classes of subgroups — too many to draw. Only its 216 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 — 4 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);(104,105,108,111)(106,109,113,116)(107,110,114,117)(112,118,119,115);(104,106)(105,109)(107,112)(108,113)(110,115)(111,116)(114,119)(117,118);(104,107)(105,110)(106,112)(108,114)(109,115)(111,117)(113,119)(116,118)3232 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C101×#13042, C101×#13044, C101×#13344, … |
| maximal normal subgroup of | C101×#13042, C101×#13044, C101×#13344, … |
| maximal quotient of | C2×C101×#1607, C2×C101×#1613, C2×C101×#1608, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12928, 1756);