Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #33982 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C17 · Schur multiplier (H2) = C25 |
| direct product | C17×#33984 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | C22×D16 |
| commutator, Frattini, cosocle | C16 | C24×C17 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ #41×C17 ◁ C17×#131 ◁ C17×#504 ◁ G | |
452 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 452 conjugacy classes of subgroups — too many to draw. Only its 202 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 145 |
| transitive degree | μt(G) = 2176 |
A minimal faithful permutation representation — 4 generators on 145 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,25,31)(20,32,37,74)(21,27,38,59)(22,42,47,71)(23,48,51,104)(24,52,50,30)(26,54,58,34)(28,63,66,45)(29,67,69,44)(33,72,75,124)(35,80,81,121)(36,82,53,73)(39,86,83,123)(40,91,84,132)(41,93,85,62)(43,68,99,122)(46,102,101,96)(49,64,105,120)(55,106,107,76)(56,103,110,98)(57,79,70,97)(60,113,111,89)(61,116,112,88)(65,100,95,119)(77,128,126,145)(78,129,94,125)(87,117,130,143)(90,139,131,135)(92,114,133,142)(108,140,141,137)(109,127,118,136)(115,138,134,144);(18,20,25,37)(19,26,31,58)(21,33,38,75)(22,43,47,99)(23,49,51,105)(24,53,50,36)(27,55,59,107)(28,64,66,120)(29,68,69,122)(30,70,52,57)(32,71,74,42)(34,63,54,45)(35,67,81,44)(39,87,83,130)(40,92,84,133)(41,94,85,78)(46,103,101,98)(48,56,104,110)(60,114,111,142)(61,117,112,143)(62,118,93,109)(65,121,95,80)(72,123,124,86)(73,96,82,102)(76,113,106,89)(77,116,126,88)(79,100,97,119)(90,140,131,137)(91,108,132,141)(115,145,134,128)(125,135,129,139)(127,138,136,144);(18,21)(19,27)(20,33)(22,39)(23,40)(24,41)(25,38)(26,55)(28,60)(29,61)(30,62)(31,59)(32,72)(34,76)(35,77)(36,78)(37,75)(42,86)(43,87)(44,88)(45,89)(46,90)(47,83)(48,91)(49,92)(50,85)(51,84)(52,93)(53,94)(54,106)(56,108)(57,109)(58,107)(63,113)(64,114)(65,115)(66,111)(67,116)(68,117)(69,112)(70,118)(71,123)(73,125)(74,124)(79,127)(80,128)(81,126)(82,129)(95,134)(96,135)(97,136)(98,137)(99,130)(100,138)(101,131)(102,139)(103,140)(104,132)(105,133)(110,141)(119,144)(120,142)(121,145)(122,143);(21,38)(27,59)(33,75)(39,83)(40,84)(41,85)(55,107)(60,111)(61,112)(62,93)(72,124)(76,106)(77,126)(78,94)(86,123)(87,130)(88,116)(89,113)(90,131)(91,132)(92,133)(108,141)(109,118)(114,142)(115,134)(117,143)(125,129)(127,136)(128,145)(135,139)(137,140)(138,144)884 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 26972);