Finite Group DatabasebetaFinite Group Databasebeta| order | 27904 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C109 · Schur multiplier (H2) = C25 |
| direct product | C109×#33984 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C109 | C22×D16 |
| commutator, Frattini, cosocle | C16 | C24×C109 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C109 ◁ #41×C109 ◁ #131×C109 ◁ #504×C109 ◁ G | |
452 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 452 conjugacy classes of subgroups — too many to draw. Only its 202 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 237 |
| transitive degree | μt(G) = 13952 |
A minimal faithful permutation representation — 4 generators on 237 points:
(1,2,8,14)(3,15,20,57)(4,10,21,42)(5,25,30,54)(6,31,34,87)(7,35,33,13)(9,37,41,17)(11,46,49,28)(12,50,52,27)(16,55,58,107)(18,63,64,104)(19,65,36,56)(22,69,66,106)(23,74,67,115)(24,76,68,45)(26,51,82,105)(29,85,84,79)(32,47,88,103)(38,89,90,59)(39,86,93,81)(40,62,53,80)(43,96,94,72)(44,99,95,71)(48,83,78,102)(60,111,109,128)(61,112,77,108)(70,100,113,126)(73,122,114,118)(75,97,116,125)(91,123,124,120)(92,110,101,119)(98,121,117,127)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237);(1,3,8,20)(2,9,14,41)(4,16,21,58)(5,26,30,82)(6,32,34,88)(7,36,33,19)(10,38,42,90)(11,47,49,103)(12,51,52,105)(13,53,35,40)(15,54,57,25)(17,46,37,28)(18,50,64,27)(22,70,66,113)(23,75,67,116)(24,77,68,61)(29,86,84,81)(31,39,87,93)(43,97,94,125)(44,100,95,126)(45,101,76,92)(48,104,78,63)(55,106,107,69)(56,79,65,85)(59,96,89,72)(60,99,109,71)(62,83,80,102)(73,123,114,120)(74,91,115,124)(98,128,117,111)(108,118,112,122)(110,121,119,127);(1,4)(2,10)(3,16)(5,22)(6,23)(7,24)(8,21)(9,38)(11,43)(12,44)(13,45)(14,42)(15,55)(17,59)(18,60)(19,61)(20,58)(25,69)(26,70)(27,71)(28,72)(29,73)(30,66)(31,74)(32,75)(33,68)(34,67)(35,76)(36,77)(37,89)(39,91)(40,92)(41,90)(46,96)(47,97)(48,98)(49,94)(50,99)(51,100)(52,95)(53,101)(54,106)(56,108)(57,107)(62,110)(63,111)(64,109)(65,112)(78,117)(79,118)(80,119)(81,120)(82,113)(83,121)(84,114)(85,122)(86,123)(87,115)(88,116)(93,124)(102,127)(103,125)(104,128)(105,126);(4,21)(10,42)(16,58)(22,66)(23,67)(24,68)(38,90)(43,94)(44,95)(45,76)(55,107)(59,89)(60,109)(61,77)(69,106)(70,113)(71,99)(72,96)(73,114)(74,115)(75,116)(91,124)(92,101)(97,125)(98,117)(100,126)(108,112)(110,119)(111,128)(118,122)(120,123)(121,127)5668 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27904, 26972);