Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13669 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C17 · Schur multiplier (H2) = C23 |
| direct product | C17×#13672 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | #2184 |
| commutator, Frattini, cosocle | C2×C16 | C23×C17 |
| prime radical | C17×#3417 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ C2×C4×C17 ◁ C17×#131 ◁ C17×#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) = 145 |
| transitive degree | μt(G) = 2176 |
A minimal faithful permutation representation — 3 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)(20,32,38,76)(21,39,44,64)(22,45,50,104)(23,51,52,29)(24,53,55,111)(26,56,61,34)(27,43,65,41)(28,66,69,49)(30,70,72,47)(33,77,81,131)(35,75,37,74)(36,86,87,128)(40,91,97,139)(42,63,100,125)(46,71,107,118)(48,109,110,102)(54,67,85,129)(57,114,116,134)(58,82,60,83)(59,105,119,106)(62,120,123,142)(68,108,103,127)(73,84,130,79)(78,88,113,117)(80,135,136,121)(89,126,122,95)(90,98,112,93)(92,132,141,115)(94,101,96,138)(99,133,137,145)(124,143,144,140);(18,20)(19,26)(21,33)(22,46)(23,35)(24,54)(25,38)(27,57)(28,67)(29,58)(30,71)(31,61)(32,45)(34,49)(36,47)(37,52)(39,88)(40,92)(41,79)(42,99)(43,84)(44,81)(48,105)(50,107)(51,60)(53,59)(55,85)(56,66)(62,121)(63,124)(64,117)(65,116)(68,128)(69,129)(70,87)(72,118)(73,101)(74,102)(75,109)(76,104)(77,91)(78,95)(80,93)(82,108)(83,127)(86,103)(89,133)(90,132)(94,140)(96,143)(97,141)(98,136)(100,137)(106,110)(111,119)(112,115)(113,126)(114,120)(122,145)(123,135)(125,144)(130,138)(131,139)(134,142);(18,21,25,44)(19,27,31,65)(20,33,38,81)(22,40,50,97)(23,43,52,41)(24,42,55,100)(26,57,61,116)(28,62,69,123)(29,64,51,39)(30,63,72,125)(32,73,76,130)(34,78,56,113)(35,84,37,79)(36,80,87,136)(45,101,104,138)(46,92,107,141)(47,93,70,98)(48,96,110,94)(49,95,66,126)(53,112,111,90)(54,99,85,137)(58,117,60,88)(59,115,119,132)(67,121,129,135)(68,122,103,89)(71,124,118,144)(74,131,75,77)(82,114,83,134)(86,133,128,145)(91,102,139,109)(105,143,106,140)(108,120,127,142)629 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 6660);