Finite Group DatabasebetaFinite Group Databasebeta| order | 21376 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C167 · Schur multiplier (H2) = C23 |
| direct product | #2117×C167 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C167 | C2×D4 |
| commutator | C4 | C22×C8×C167 |
| Frattini, cosocle | C2×C8 | C23×C167 |
| socle | C2×C167 | C2×C22⋊C8 |
| prime radical | C2×#41×C167 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C167 ◁ C22×C8×C167 ◁ G | |
220 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 199 |
| transitive degree | μt(G) = 5344 |
A minimal faithful permutation representation — 3 generators on 199 points:
(1,2,5,11,6,12,20,29,7,9,21,25,4,10,18,27)(3,13,15,30,14,23,31,28,16,26,19,32,17,8,22,24)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199);(1,3)(2,8)(4,17)(5,19)(6,14)(7,16)(9,23)(10,26)(11,28)(12,13)(15,21)(18,31)(20,22)(24,25)(27,30)(29,32);(2,9)(8,23)(10,12)(11,25)(13,26)(24,28)(27,29)(30,32)7348 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21376, 849);