Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | #577×C97 ⋊ C2 |
| Sylow | C97 ⋊ #3584 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×D4×D97 |
| commutator, cosocle | C4×C97 | C25 |
| Frattini | C4 | C24×D97 |
| Fitting | #577×C97 | C2 |
| socle | C2×C97 | C23×D4 |
| hypercenter | #577 | D97 |
| hypocenter | C97 | #3584 |
| LCS | G ▷ C4×C97 ▷ C2×C97 ▷ C97 | |
| UCS | C1 ◁ C2 ◁ #41 ◁ #577 | |
1424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 3104 |
A minimal faithful permutation representation — 5 generators on 129 points:
(1,2)(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)(98,102,104,103)(99,107,108,116)(100,111,112,117)(101,114,113,115)(105,119,120,109)(106,122,121,126)(110,125,124,127)(118,129,128,123);(101,113)(106,121)(110,124)(114,115)(118,128)(122,126)(123,129)(125,127);(98,101)(99,106)(100,110)(102,114)(103,115)(104,113)(105,118)(107,122)(108,121)(109,123)(111,125)(112,124)(116,126)(117,127)(119,129)(120,128);(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)(99,105)(101,110)(102,111)(103,117)(104,112)(106,118)(107,119)(108,120)(109,116)(113,124)(114,125)(115,127)(121,128)(122,129)(123,126);(98,99)(100,109)(101,106)(102,107)(103,116)(104,108)(105,111)(110,123)(112,119)(113,121)(114,122)(115,126)(117,120)(118,125)(124,129)(127,128)Polycyclic presentation:
2^7,97|e^2=g,f^2=g,b^a=bf,f^a=fg,f^b=fg,h^b=h^96,d^c=dg,h^e=h^961100 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 19282);