Finite Group DatabasebetaFinite Group Databasebeta| order | 2816 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #13742 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#13743 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | C2×D32 |
| commutator, Frattini, cosocle | C32 | C23×C11 |
| prime radical | C11×#2262 | C2 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C2×C4×C11 ◁ C2×C8×C11 ◁ C2×C11×C16 ◁ C2×C11×C32 ◁ G | |
126 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 126 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) = 139 |
| transitive degree | μt(G) = 1408 |
A minimal faithful permutation representation — 3 generators on 139 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,26)(15,32)(16,39)(17,44)(18,48)(19,21)(20,27)(22,35)(23,34)(24,41)(25,46)(28,72)(29,76)(30,79)(31,65)(33,60)(36,92)(37,96)(38,64)(40,57)(42,104)(43,98)(45,66)(47,106)(49,67)(50,71)(51,84)(52,83)(53,109)(54,68)(55,73)(56,77)(58,118)(59,120)(61,124)(62,69)(63,70)(74,135)(75,134)(78,136)(80,87)(81,90)(82,94)(85,125)(86,115)(88,100)(89,99)(91,108)(93,122)(95,129)(97,123)(101,119)(102,121)(103,113)(105,117)(107,128)(110,138)(111,137)(112,139)(114,132)(116,133)(126,131)(127,130);(12,14,19,31)(13,20,21,50)(15,33,38,86)(16,40,43,103)(17,45,47,107)(18,49,46,30)(22,57,54,113)(23,60,55,115)(24,62,56,116)(25,63,48,53)(26,64,65,32)(27,68,71,35)(28,73,75,34)(29,77,78,41)(36,93,95,139)(37,97,94,85)(39,51,98,110)(42,105,88,102)(44,52,106,111)(58,119,114,135)(59,121,108,117)(61,125,99,123)(66,129,128,92)(67,82,79,96)(69,132,133,118)(70,91,109,120)(72,126,134,80)(74,89,101,124)(76,127,136,81)(83,131,137,87)(84,130,138,90)(100,122,104,112);(13,21)(20,50)(22,54)(23,55)(24,56)(25,48)(26,65)(32,64)(39,98)(44,106)(51,110)(52,111)(53,63)(57,113)(58,114)(59,108)(60,115)(61,99)(62,116)(66,128)(67,79)(72,134)(76,136)(80,126)(81,127)(82,96)(92,129)(100,104)(112,122)(117,121)(119,135)(123,125)506 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2816, 6731);