Finite Group DatabasebetaFinite Group Databasebeta| order | 12928 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C101 · Schur multiplier (H2) = C22×C4 |
| direct product | C4×C4⋊C8×C101 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42×C101 | C22 |
| commutator | C2 | C2×C4×C8×C101 |
| Frattini, cosocle | C22×C4 | C23×C101 |
| socle, prime radical | C23×C101 | C22×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42×C101 ◁ G | |
316 conjugacy classes of subgroups; only the 240 normal subgroups are shown.
This group has 316 conjugacy classes of subgroups — too many to draw. Only its 240 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 117 |
| transitive degree | μt(G) = 12928 |
A minimal faithful permutation representation — 3 generators on 117 points:
(1,2,3,4)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117);(6,8)(9,10,11,13,12,14,15,16);(5,6,7,8)8080 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C101×#7015, C101×#7022, C101×#7023, … |
| maximal normal subgroup of | C101×#7015, C101×#7022, C101×#7023, … |
| maximal quotient of | C8×C4⋊C8×C101, C101×#8260, C101×#8262, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12928, 498);