Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C109 · Schur multiplier (H2) = C23 |
| direct product | C109×#12942 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C109 | #447 |
| commutator | C2×C8 | C22×C4×C109 |
| Frattini, cosocle | C22×C8 | C23×C109 |
| socle | C2×C109 | #1862 |
| prime radical | C2×#135×C109 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C109 ◁ C22×C4×C109 ◁ #405×C109 ◁ G | |
380 conjugacy classes of subgroups.
| minimal degree | μ(G) = 173 |
| transitive degree | μt(G) = 6976 |
A minimal faithful permutation representation — 3 generators on 173 points:
(1,2,7,14,9,15,8,30)(3,16,20,34,21,47,31,43)(4,22,25,39,26,59,32,58)(5,13,28,44,29,45,6,12)(10,27,35,55,37,19,36,18)(11,38,40,60,42,23,41,24)(17,48,51,33,53,62,52,63)(46,56,64,54,61,49,57,50)(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);(1,3,7,20,9,21,8,31)(2,10,14,35,15,37,30,36)(4,17,25,51,26,53,32,52)(5,27,28,55,29,19,6,18)(11,33,40,62,42,63,41,48)(12,43,13,16,44,34,45,47)(22,57,39,46,59,64,58,61)(23,54,24,49,38,50,60,56);(1,4)(2,11)(3,17)(5,24)(6,23)(7,25)(8,32)(9,26)(10,33)(12,22)(13,39)(14,40)(15,42)(16,46)(18,54)(19,56)(20,51)(21,53)(27,49)(28,38)(29,60)(30,41)(31,52)(34,64)(35,62)(36,48)(37,63)(43,57)(44,59)(45,58)(47,61)(50,55)5014 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 5930);