Finite Group DatabasebetaFinite Group Databasebeta| order | 24704 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C193 · Schur multiplier (H2) = C2 |
| direct product | #1417×C193 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C193 | D16 |
| commutator | C16 | C2×C4×C193 |
| Frattini, cosocle | C2×C16 | C22×C193 |
| socle | C2×C193 | #357 |
| prime radical | #508×C193 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C193 ◁ C2×C4×C193 ◁ C2×C8×C193 ◁ C2×C16×C193 ◁ G | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 257 |
| transitive degree | μt(G) = 12352 |
A minimal faithful permutation representation — 2 generators on 257 points:
(1,2,5,10,7,13,26,12)(3,14,16,43,19,44,18,50)(4,20,23,51,25,53,58,52)(6,27,22,11,29,39,57,41)(8,30,31,46,33,45,42,15)(9,34,36,59,38,61,37,24)(17,48,47,62,49,63,28,35)(21,40,54,60,56,64,55,32)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257);(1,3)(2,8)(4,21)(5,16)(6,28)(7,19)(9,35)(10,31)(11,40)(12,42)(13,33)(14,20)(15,24)(17,22)(18,26)(23,54)(25,56)(27,32)(29,47)(30,34)(36,48)(37,63)(38,62)(39,60)(41,64)(43,51)(44,53)(45,61)(46,59)(49,57)(50,52)(55,58)7334 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(24704, 149);