Finite Group DatabasebetaFinite Group Databasebeta| order | 21376 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C167 · Schur multiplier (H2) = C22 |
| direct product | #1658×C167 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C167 | #120 |
| commutator, Frattini, cosocle | C42 | C23×C167 |
| prime radical | #127×C167 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C167 ◁ C42×C167 ◁ G | |
182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 199 |
| transitive degree | μt(G) = 10688 |
A minimal faithful permutation representation — 3 generators on 199 points:
(2,7)(3,10)(4,5)(9,12)(11,13)(14,15)(18,22)(20,27)(23,31)(24,28)(25,29)(30,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199);(1,2)(3,8)(4,12)(5,7)(6,9)(10,15)(11,16)(13,14)(17,18,21,24)(19,23,26,32)(20,28,27,22)(25,30,29,31);(1,3,6,11)(2,8,9,16)(4,10,5,13)(7,14,12,15)(17,19,21,26)(18,23,24,32)(20,29,27,25)(22,30,28,31)3340 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21376, 390);