Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | #459×C97 ⋊ C2 |
| Sylow | C97 ⋊ #1619 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #15690778 |
| commutator, cosocle | C42×C97 | C23 |
| Frattini | C42 | C22×D97 |
| Fitting | #459×C97 | C2 |
| socle | C22×C97 | #120 |
| hypercenter | #459 | D97 |
| hypocenter | C97 | #1619 |
| LCS | G ▷ C42×C97 ▷ C22×C97 ▷ C97 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #459 | |
336 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 3104 |
A minimal faithful permutation representation — 3 generators on 113 points:
(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(107,112)(108,109)(111,113);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)(86,87)(88,89)(90,91)(92,93)(94,95)(96,97)(99,102)(100,105)(103,104)(107,111)(108,109)(112,113);(98,99)(100,102)(101,104)(103,105)(106,107)(108,111)(109,112)(110,113)Polycyclic presentation:
2^7,97|d^2=f,e^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,d^b=df,e^b=eg,h^b=h^96,d^c=dg,e^c=eg938 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 3563);