Finite Group DatabasebetaFinite Group Databasebeta| order | 30592 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C239 · Schur multiplier (H2) = C22 |
| direct product | #1617×C239 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C239 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C23×C239 |
| prime radical | C23×C239 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C239 ◁ C22×C4×C239 ◁ G | |
202 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 202 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 271 |
| transitive degree | μt(G) = 7648 |
A minimal faithful permutation representation — 3 generators on 271 points:
(1,2,7,22)(3,12,14,29)(4,17,23,19)(5,10,6,11)(8,24,25,13)(9,21,32,20)(15,28,16,30)(18,26,31,27)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271);(1,3,5,15)(2,8,10,26)(4,18,20,13)(6,16,7,14)(9,28,17,12)(11,27,22,25)(19,29,32,30)(21,24,23,31);(1,2,6,11)(3,8,16,27)(4,19,21,9)(5,10,7,22)(12,13,30,31)(14,25,15,26)(17,20,32,23)(18,29,24,28)5497 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(30592, 349);