Finite Group DatabasebetaFinite Group Databasebeta| order | 36224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C283 · Schur multiplier (H2) = C22 |
| direct product | #2170×C283 |
| subgroup | G/H | |
|---|---|---|
| center | C16×C283 | D4 |
| commutator | C4 | C22×C8×C283 |
| Frattini, cosocle | C2×C8 | C23×C283 |
| socle | C2×C283 | C8×D4 |
| prime radical | #135×C283 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16×C283 ◁ C2×C16×C283 ◁ G | |
146 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 315 |
| transitive degree | μt(G) = 9056 |
A minimal faithful permutation representation — 3 generators on 315 points:
(1,2,6,10,5,11,20,28,7,12,22,29,21,9,25,27)(3,13,15,18,16,24,30,19,17,26,31,32,14,8,23,4)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315);(1,3)(2,8)(4,10)(5,16)(6,15)(7,17)(9,26)(11,13)(12,24)(14,21)(18,28)(19,29)(20,30)(22,31)(23,25)(27,32);(1,4,6,13,5,18,20,24,7,19,22,26,21,32,25,8)(2,3,10,15,11,16,28,30,12,17,29,31,9,14,27,23)15848 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(36224, 902);