Finite Group DatabasebetaFinite Group Databasebeta| order | 21376 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C167 · Schur multiplier (H2) = C2 |
| direct product | #1323×C167 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C167 | C22⋊C4 |
| commutator | C2×C4 | C2×C8×C167 |
| Frattini, cosocle | C22×C8 | C22×C167 |
| socle | C22×C167 | C22⋊C8 |
| prime radical | C23×C167 | M16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C167 ◁ C22×C8×C167 ◁ G | |
94 conjugacy classes of subgroups.
| minimal degree | μ(G) = 207 |
| transitive degree | μt(G) = 10688 |
A minimal faithful permutation representation — 2 generators on 207 points:
(1,2)(3,7,8,5)(4,6)(9,10,13,18,14,19,28,36,15,20,29,37,12,17,26,34)(11,21,23,38,22,32,39,35,24,33,27,40,25,16,30,31)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,25)(13,27)(14,22)(15,24)(17,33)(18,35)(19,21)(20,32)(23,29)(26,39)(28,30)(31,37)(34,38)(36,40)6346 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21376, 55);