Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | #95×C97 ⋊ C4 |
| Sylow | C97 ⋊ #1443 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×#87604 |
| commutator, socle | C23×C97 | C22×C4 |
| Fitting | #95×C97 | C4 |
| prime radical, cosocle | C23×D97 | C23 |
| hypercenter | #95 | #87604 |
| hypocenter | C97 | #1443 |
| LCS | G ▷ C23×C97 ▷ C97 | |
| UCS | C1 ◁ C23 ◁ #95 | |
476 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 476 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 3104 |
A minimal faithful permutation representation — 3 generators on 109 points:
(1,2,5,10,19,32,49,70,64,45,62,80,72,47,67,85,92,83,63,42,25,12,3,7,14,23,36,53,74,60,43,28,41,58,78,68,87,94,96,90,81,59,38,21,8,16,27,40,57,77,56,39,26,15,24,37,54,75,89,95,97,93,86,79,55,34,17,29,44,61,73,52,35,22,13,6,11,20,33,50,71,88,91,84,66,82,76,51,30,46,65,69,48,31,18,9,4)(99,101)(102,104,105,103)(106,107)(108,109);(98,99)(100,101)(102,103)(104,105)(106,107,108,109);(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)(103,104)(106,107,108,109)Polycyclic presentation:
2^7,97|a^2=g,b^2=de,c^2=d,b^a=bd,c^a=ce,h^a=h^22,c^b=cf,h^g=h^96524 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 19697);