Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13669 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#13672 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | #2184 |
| commutator, Frattini, cosocle | C2×C16 | C23×C7 |
| prime radical | C7×#3417 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ C2×C4×C7 ◁ C7×#131 ◁ C7×#504 ◁ G | |
274 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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)(8,9,15,21)(10,22,28,66)(11,29,34,54)(12,35,40,94)(13,41,42,19)(14,43,45,101)(16,46,51,24)(17,33,55,31)(18,56,59,39)(20,60,62,37)(23,67,71,121)(25,65,27,64)(26,76,77,118)(30,81,87,129)(32,53,90,115)(36,61,97,108)(38,99,100,92)(44,57,75,119)(47,104,106,124)(48,72,50,73)(49,95,109,96)(52,110,113,132)(58,98,93,117)(63,74,120,69)(68,78,103,107)(70,125,126,111)(79,116,112,85)(80,88,102,83)(82,122,131,105)(84,91,86,128)(89,123,127,135)(114,133,134,130);(8,10)(9,16)(11,23)(12,36)(13,25)(14,44)(15,28)(17,47)(18,57)(19,48)(20,61)(21,51)(22,35)(24,39)(26,37)(27,42)(29,78)(30,82)(31,69)(32,89)(33,74)(34,71)(38,95)(40,97)(41,50)(43,49)(45,75)(46,56)(52,111)(53,114)(54,107)(55,106)(58,118)(59,119)(60,77)(62,108)(63,91)(64,92)(65,99)(66,94)(67,81)(68,85)(70,83)(72,98)(73,117)(76,93)(79,123)(80,122)(84,130)(86,133)(87,131)(88,126)(90,127)(96,100)(101,109)(102,105)(103,116)(104,110)(112,135)(113,125)(115,134)(120,128)(121,129)(124,132);(8,11,15,34)(9,17,21,55)(10,23,28,71)(12,30,40,87)(13,33,42,31)(14,32,45,90)(16,47,51,106)(18,52,59,113)(19,54,41,29)(20,53,62,115)(22,63,66,120)(24,68,46,103)(25,74,27,69)(26,70,77,126)(35,91,94,128)(36,82,97,131)(37,83,60,88)(38,86,100,84)(39,85,56,116)(43,102,101,80)(44,89,75,127)(48,107,50,78)(49,105,109,122)(57,111,119,125)(58,112,93,79)(61,114,108,134)(64,121,65,67)(72,104,73,124)(76,123,118,135)(81,92,129,99)(95,133,96,130)(98,110,117,132)259 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6660);