Finite Group DatabasebetaFinite Group Databasebeta| order | 21376 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C167 · Schur multiplier (H2) = C22 |
| direct product | #1385×C167 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C167 | #106 |
| commutator | C8 | C42×C167 |
| Frattini, cosocle | C22×C8 | C22×C167 |
| socle | C2×C167 | #340 |
| prime radical | C23×C167 | Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C167 ◁ C22×C4×C167 ◁ C22×C8×C167 ◁ G | |
124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 199 |
| transitive degree | μt(G) = 5344 |
A minimal faithful permutation representation — 2 generators on 199 points:
(1,2,5,11,8,13,7,23)(3,14,16,29,18,30,21,25)(4,12,6,22,20,28,19,10)(9,17,24,15,27,31,26,32)(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);(1,3,6,17,5,21,20,32,8,18,19,31,7,16,4,15)(2,9,12,25,23,24,10,14,13,27,28,29,11,26,22,30)5344 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21376, 117);