Finite Group DatabasebetaFinite Group Databasebeta| order | 21376 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C167 · Schur multiplier (H2) = C23 |
| direct product | #1813×C167 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C167 | #117 |
| commutator | C2×C4 | C22×C4×C167 |
| Frattini, cosocle | C22×C4 | C23×C167 |
| socle | C2×C167 | #380 |
| prime radical | C2×#41×C167 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C167 ◁ C2×M16×C167 ◁ G | |
252 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 252 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:
(1,2,6,10,7,11,5,20)(3,12,15,23,16,27,14,8)(4,17,18,24,19,9,21,25)(13,28,29,32,30,26,31,22)(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);(1,3,5,14,7,16,6,15)(2,8,10,12,11,23,20,27)(4,13,18,29,19,30,21,31)(9,22,24,26,17,32,25,28);(1,4)(2,9)(3,13)(5,18)(6,21)(7,19)(8,22)(10,24)(11,17)(12,26)(14,29)(15,31)(16,30)(20,25)(23,32)(27,28)5344 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21376, 545);