Finite Group DatabasebetaFinite Group Databasebeta| order | 38656 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C151 · Schur multiplier (H2) = C2 |
| direct product | C151×#12466 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C151 | #1677 |
| commutator, Frattini, cosocle | C4×C8 | C23×C151 |
| prime radical | #135×C151 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C151 ◁ C2×C4×C151 ◁ C4×C8×C151 ◁ G | |
186 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 215 |
| transitive degree | μt(G) = 9664 |
A minimal faithful permutation representation — 3 generators on 215 points:
(1,2,9,16)(3,17,24,49)(4,25,29,41)(5,13,30,35)(6,12,18,44)(7,15,8,14)(10,34,40,20)(11,26,43,27)(19,51,54,63)(21,38,31,39)(22,37,23,45)(28,58,33,46)(32,47,50,42)(36,57,61,53)(48,55,60,52)(56,62,59,64)(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);(1,3,18,21,7,22,5,20,9,24,6,31,8,23,30,34)(2,10,35,37,14,38,12,17,16,40,13,45,15,39,44,49)(4,19,50,56,28,53,27,55,29,54,32,59,33,57,26,52)(11,36,58,62,42,51,25,48,43,61,46,64,47,63,41,60);(1,4)(2,11)(3,19)(5,27)(6,32)(7,28)(8,33)(9,29)(10,36)(12,25)(13,46)(14,42)(15,47)(16,43)(17,48)(18,50)(20,55)(21,56)(22,53)(23,57)(24,54)(26,30)(31,59)(34,52)(35,58)(37,62)(38,51)(39,63)(40,61)(41,44)(45,64)(49,60)4681 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38656, 5454);