Finite Group DatabasebetaFinite Group Databasebeta| order | 23168 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C181 · Schur multiplier (H2) = C1 |
| direct product | #1426×C181 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C181 | #367 |
| commutator | C16 | C2×C4×C181 |
| Frattini, cosocle | C2×C16 | C22×C181 |
| prime radical | C22×C181 | SD32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C181 ◁ C2×C4×C181 ◁ C2×C8×C181 ◁ C2×C16×C181 ◁ G | |
52 conjugacy classes of subgroups.
| minimal degree | μ(G) = 213 |
| transitive degree | μt(G) = 5792 |
A minimal faithful permutation representation — 2 generators on 213 points:
(2,7,9,20)(3,12)(4,16)(5,17)(8,23)(10,24,27,32)(11,30,28,22)(13,18)(14,26)(15,25)(19,31,29,21)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213);(1,2,8,24,4,10,3,7,17,28,12,21,13,19,23,30,6,9,26,32,18,27,15,20,5,11,25,31,16,29,14,22)4706 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(23168, 158);