Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C97 ⋊ #1917 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×#87604 |
| commutator, socle, prime radical | C23×C97 | C22×C4 |
| Fitting | #95×C97 | C4 |
| cosocle | C22×Dic97 | C23 |
| hypercenter | #95 | #87604 |
| hypocenter | C97 | #1917 |
| LCS | G ▷ C23×C97 ▷ C97 | |
| UCS | C1 ◁ C23 ◁ #95 | |
228 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 117 |
| transitive degree | μt(G) = 12416 |
A minimal faithful permutation representation — 3 generators on 117 points:
(1,2)(3,11)(4,5)(6,7)(8,24)(9,10)(12,20)(13,14)(15,16)(17,41)(18,19)(21,37)(22,23)(25,33)(26,27)(28,29)(30,62)(31,32)(34,58)(35,36)(38,54)(39,40)(42,50)(43,44)(45,46)(47,82)(48,49)(51,80)(52,53)(55,78)(56,57)(59,75)(60,61)(63,71)(64,65)(66,67)(68,79)(69,70)(72,76)(73,74)(81,89)(83,88)(84,85)(86,87)(90,95)(91,92)(93,94)(96,97)(99,101)(102,104,105,108)(103,107,106,109)(110,113,114,115)(111,112,116,117);(2,3,4,6)(5,8,9,15)(7,12,13,11)(10,17,18,28)(14,21,22,24)(16,25,26,20)(19,30,31,45)(23,34,35,41)(27,38,39,37)(29,42,43,33)(32,47,48,66)(36,51,52,62)(40,55,56,58)(44,59,60,54)(46,63,64,50)(49,68,69,86)(53,72,73,82)(57,76,77,80)(61,79,74,78)(65,81,70,75)(67,83,84,71)(85,90,91,89)(87,92,93,88)(94,96,97,95)(103,106)(107,109)(110,112,115,116,114,117,113,111);(2,3,4,6)(5,8,9,15)(7,12,13,11)(10,17,18,28)(14,21,22,24)(16,25,26,20)(19,30,31,45)(23,34,35,41)(27,38,39,37)(29,42,43,33)(32,47,48,66)(36,51,52,62)(40,55,56,58)(44,59,60,54)(46,63,64,50)(49,68,69,86)(53,72,73,82)(57,76,77,80)(61,79,74,78)(65,81,70,75)(67,83,84,71)(85,90,91,89)(87,92,93,88)(94,96,97,95)(98,99)(100,101)(102,103)(104,107)(105,106)(108,109)(110,111,115,112,114,116,113,117)Polycyclic presentation:
2^7,97|a^2=f,b^2=f,c^2=d,f^2=g,b^a=bd,c^a=ce,h^a=h^22,c^b=cg,h^b=h^22,h^c=h^96,h^f=h^96524 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 20568);