Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
No aliases for this group.
| Fitting | #585×C97 ⋊ C2 |
| Sylow | C97 ⋊ 2−1+6 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C25×D97 |
| commutator, socle, cosocle | C2×C97 | C26 |
| Fitting | #585×C97 | C2 |
| hypercenter | #585 | D97 |
| hypocenter | C97 | 2−1+6 |
| LCS | G ▷ C2×C97 ▷ C97 | |
| UCS | C1 ◁ C2 ◁ #585 | |
5796 conjugacy classes of subgroups; only the 3201 normal subgroups are shown.
This group has 5796 conjugacy classes of subgroups — too many to draw. Only its 3201 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 — 6 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)(99,107)(104,108)(105,119)(106,121)(116,122)(117,123)(118,120)(128,129);(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)(100,111)(104,108)(109,125)(110,127)(116,122)(117,123)(124,126)(128,129);(98,102,103,115)(99,106,107,121)(100,110,111,127)(101,112,113,114)(104,117,108,123)(105,118,119,120)(109,124,125,126)(116,128,122,129);(98,101,103,113)(99,105,107,119)(100,109,111,125)(102,114,115,112)(104,116,108,122)(106,120,121,118)(110,126,127,124)(117,129,123,128);(98,100)(99,104)(101,109)(102,110)(103,111)(105,116)(106,117)(107,108)(112,124)(113,125)(114,126)(115,127)(118,128)(119,122)(120,129)(121,123);(98,99)(100,108)(101,105)(102,106)(103,107)(104,111)(109,122)(110,123)(112,118)(113,119)(114,120)(115,121)(116,125)(117,127)(124,129)(126,128)Polycyclic presentation:
2^7,97|c^2=g,d^2=g,b^a=bg,f^a=fg,e^b=eg,d^c=dg,h^e=h^96,h^f=h^961697 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 19323);