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 | C23 | C22×#87604 |
| commutator, socle, prime radical | C23×C97 | C22×C4 |
| Fitting | #95×C97 | C4 |
| cosocle | C22×Dic97 | C23 |
| hypercenter | #95 | #87604 |
| hypocenter | C97 | #1840 |
| LCS | G ▷ C23×C97 ▷ C97 | |
| UCS | C1 ◁ C23 ◁ #95 | |
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:
(1,2,5,10,19,32,49,70,64,45,62,80,72,47,67,85,92,83,63,42,25,12,3,7,14,23,36,53,74,60,43,28,41,58,78,68,87,94,96,90,81,59,38,21,8,16,27,40,57,77,56,39,26,15,24,37,54,75,89,95,97,93,86,79,55,34,17,29,44,61,73,52,35,22,13,6,11,20,33,50,71,88,91,84,66,82,76,51,30,46,65,69,48,31,18,9,4)(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);(2,3,4,6)(5,8,9,15)(7,12,13,11)(10,17,18,28)(14,21,22,24)(16,25,26,20)(19,30,31,45)(23,34,35,41)(27,38,39,37)(29,42,43,33)(32,47,48,66)(36,51,52,62)(40,55,56,58)(44,59,60,54)(46,63,64,50)(49,68,69,86)(53,72,73,82)(57,76,77,80)(61,79,74,78)(65,81,70,75)(67,83,84,71)(85,90,91,89)(87,92,93,88)(94,96,97,95)(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^22,c^b=cg,h^f=h^96524 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 20439);