Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C97 ⋊ #1840 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22×C4 | C22×D97 |
| commutator, socle, prime radical | C23×C97 | C22×C4 |
| Fitting | #376×C97 | C2 |
| cosocle | C22×C4×C97 | C23 |
| hypercenter | #376 | D97 |
| hypocenter | C97 | #1840 |
| LCS | G ▷ C23×C97 ▷ C97 | |
| UCS | C1 ◁ C22×C4 ◁ #376 | |
196 conjugacy classes of subgroups.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 12416 |
A minimal faithful permutation representation — 3 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)(99,103)(102,105)(106,108,109,113)(107,111,110,112)(116,122)(119,125)(123,128)(126,129);(98,100,101,104)(99,102,103,105)(107,110)(111,112)(114,116)(115,119)(117,123)(118,122)(120,126)(121,125)(124,128)(127,129);(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)(98,99)(100,102)(101,103)(104,105)(106,107)(108,112)(109,110)(111,113)(114,115,117,120,118,121,124,127)(116,119,123,126,122,125,128,129)Polycyclic presentation:
2^7,97|a^2=f,b^2=de,c^2=d,f^2=g,b^a=bd,c^a=ce,h^a=h^96,c^b=cg1964 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 4655);