Finite Group DatabasebetaFinite Group Databasebeta| order | 38144 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16×C149 · Schur multiplier (H2) = C2 |
| direct product | C149×#7363 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C149 | C22⋊C4 |
| commutator | C2×C4 | C2×C16×C149 |
| Frattini, cosocle | C22×C16 | C22×C149 |
| socle | C22×C149 | C22⋊C16 |
| prime radical | C23×C149 | M32 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C149 ◁ C22×C16×C149 ◁ G | |
124 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 58 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 221 |
| transitive degree | μt(G) = 19072 |
A minimal faithful permutation representation — 2 generators on 221 points:
(1,2)(3,7,8,5)(4,6)(9,10,13,19,14,20,32,45,15,21,33,46,35,48,60,68,16,22,34,47,36,49,61,69,12,18,29,42,30,43,57,66)(11,23,25,50,26,51,54,70,24,40,52,44,53,64,71,67,27,41,31,58,55,65,59,72,28,17,37,38,56,39,62,63)(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);(1,3,4,8)(2,5,6,7)(9,11)(10,17)(12,28)(13,31)(14,26)(15,24)(16,27)(18,41)(19,44)(20,39)(21,23)(22,40)(25,34)(29,52)(30,56)(32,59)(33,37)(35,53)(36,55)(38,47)(42,50)(43,65)(45,67)(46,58)(48,51)(49,64)(54,61)(57,71)(60,62)(63,69)(66,70)(68,72)11324 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(38144, 351);