Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C109 · Schur multiplier (H2) = C22 |
| direct product | C109×#11647 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C109 | C2×D8 |
| commutator | C2×C8 | C22×C4×C109 |
| Frattini, cosocle | C4×C8 | C23×C109 |
| socle | C22×C109 | C2×#102 |
| prime radical | #466×C109 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C109 ◁ C2×C42×C109 ◁ C2×C4×C8×C109 ◁ G | |
310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 157 |
| transitive degree | μt(G) = 13952 |
A minimal faithful permutation representation — 3 generators on 157 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,28)(20,25)(21,33)(22,37)(23,27)(24,30)(26,35)(29,44)(31,45)(32,43)(34,39)(36,47)(38,42)(40,48)(41,46)(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);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19)(18,24)(20,29)(21,34)(22,38)(23,31)(25,39)(26,42)(27,41)(28,33)(30,35)(32,46)(36,48)(37,40)(43,44)(45,47);(3,9)(6,12)(10,15)(13,16)(17,20,23,22)(18,25,27,37)(19,29,31,38)(21,32,36,35)(24,39,41,40)(26,33,43,47)(28,44,45,42)(30,34,46,48)6322 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 4635);