Finite Group DatabasebetaFinite Group Databasebeta| order | 22912 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C179 · Schur multiplier (H2) = C2 |
| direct product | #1358×C179 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C179 | #331 |
| commutator | C42 | C2×C4×C179 |
| Frattini, cosocle | C4×C8 | C22×C179 |
| prime radical | C22×C179 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C179 ◁ C2×C4×C179 ◁ C4×C8×C179 ◁ G | |
92 conjugacy classes of subgroups.
| minimal degree | μ(G) = 211 |
| transitive degree | μt(G) = 5728 |
A minimal faithful permutation representation — 2 generators on 211 points:
(1,2,4,9,5,8,17,24,6,11,18,27,20,10,28,26)(3,12,13,29,14,23,30,32,15,22,31,25,19,7,16,21)(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);(1,3,6,15)(2,7,11,23)(4,16,18,30)(5,19,20,14)(8,22,10,12)(9,25,27,29)(13,17,31,28)(21,24,32,26)4117 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22912, 90);