Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | #568×C97 ⋊ C2 |
| Sylow | C97 ⋊ #3586 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C23×D388 |
| commutator, cosocle | C4×C97 | C25 |
| Frattini | C4 | C24×D97 |
| Fitting | #568×C97 | C2 |
| socle | C2×C97 | C23×D4 |
| hypercenter | #568 | D97 |
| hypocenter | C97 | #3586 |
| LCS | G ▷ C4×C97 ▷ C2×C97 ▷ C97 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ #568 | |
1412 conjugacy classes of subgroups; only the 545 normal subgroups are shown.
This group has 1412 conjugacy classes of subgroups — too many to draw. Only its 545 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 3104 |
A minimal faithful permutation representation — 5 generators on 129 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)(99,106)(104,118)(105,119)(107,114)(108,121)(117,128)(120,126)(122,129);(101,112)(105,119)(109,123)(113,125)(117,128)(120,126)(122,129)(124,127);(98,101)(99,105)(100,109)(102,113)(103,112)(104,117)(106,119)(107,120)(108,122)(110,124)(111,123)(114,126)(115,127)(116,125)(118,128)(121,129);(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)(98,100,103,111)(99,104,106,118)(101,109,112,123)(102,115,116,110)(105,117,119,128)(107,121,114,108)(113,127,125,124)(120,129,126,122);(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)(98,99)(100,108)(101,105)(102,114)(103,106)(104,110)(107,116)(109,122)(111,121)(112,119)(113,126)(115,118)(117,124)(120,125)(123,129)(127,128)Polycyclic presentation:
2^7,97|b^2=g,f^2=g,b^a=bf,e^a=eg,f^a=fg,h^a=h^96,f^b=fg,h^b=h^96,d^c=dg1673 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 19298);