Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C7×C32 · Schur multiplier (H2) = C22 |
| direct product | C7×#13737 |
| subgroup | G/H | |
|---|---|---|
| center | C7×C64 | C22 |
| commutator | C2 | C22×C7×C32 |
| Frattini, cosocle | C32 | C23×C7 |
| socle | C2×C7 | C22×C32 |
| prime radical | C7×#41 | C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C7×C64 ◁ G | |
128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 896 |
A minimal faithful permutation representation — 3 generators on 135 points:
(1,2,3,4,5,6,7)(8,9,11,17,12,18,27,42,13,19,28,43,31,46,62,82,14,20,29,44,32,47,63,83,34,49,65,85,68,88,102,117,15,21,30,45,33,48,64,84,35,50,66,86,69,89,103,118,36,51,67,87,70,90,104,119,71,91,105,120,106,121,127,133)(10,16,23,38,24,39,56,76,25,40,57,77,59,79,98,113,26,41,58,78,60,80,99,114,61,81,100,115,101,116,126,132,22,37,52,72,53,73,92,107,54,74,93,108,95,110,122,128,55,75,94,109,96,111,123,129,97,112,124,130,125,131,134,135);(8,10)(9,16)(11,23)(12,24)(13,25)(14,26)(15,22)(17,38)(18,39)(19,40)(20,41)(21,37)(27,56)(28,57)(29,58)(30,52)(31,59)(32,60)(33,53)(34,61)(35,54)(36,55)(42,76)(43,77)(44,78)(45,72)(46,79)(47,80)(48,73)(49,81)(50,74)(51,75)(62,98)(63,99)(64,92)(65,100)(66,93)(67,94)(68,101)(69,95)(70,96)(71,97)(82,113)(83,114)(84,107)(85,115)(86,108)(87,109)(88,116)(89,110)(90,111)(91,112)(102,126)(103,122)(104,123)(105,124)(106,125)(117,132)(118,128)(119,129)(120,130)(121,131)(127,134)(133,135);(10,22)(16,37)(23,52)(24,53)(25,54)(26,55)(38,72)(39,73)(40,74)(41,75)(56,92)(57,93)(58,94)(59,95)(60,96)(61,97)(76,107)(77,108)(78,109)(79,110)(80,111)(81,112)(98,122)(99,123)(100,124)(101,125)(113,128)(114,129)(115,130)(116,131)(126,134)(132,135)1120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6725);