Finite Group DatabasebetaFinite Group Databasebeta| order | 13184 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | #529×C103 ⋊ C2 |
| Sylow | C103 ⋊ #3531 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D103 |
| commutator, socle, cosocle | C22×C103 | C25 |
| Frattini | C22 | C24×D103 |
| Fitting | #529×C103 | C2 |
| hypercenter | #529 | D103 |
| hypocenter | C103 | #3531 |
| LCS | G ▷ C22×C103 ▷ C103 | |
| UCS | C1 ◁ C2×C4 ◁ #529 | |
930 conjugacy classes of subgroups.
| minimal degree | μ(G) = 135 |
| transitive degree | μt(G) = 6592 |
A minimal faithful permutation representation — 5 generators on 135 points:
(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98,100,102,103,101,99,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(120,123,124,131)(121,127,126,133)(122,130,129,135)(125,132,128,134);(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)(100,101)(102,103)(105,111)(109,118)(110,114)(117,119)(121,126)(122,129)(127,133)(130,135);(104,107,108,116)(105,110,111,114)(106,112,113,115)(109,117,118,119)(122,129)(125,128)(130,135)(132,134);(104,106,108,113)(105,109,111,118)(107,115,116,112)(110,119,114,117)(120,122)(121,125)(123,130)(124,129)(126,128)(127,132)(131,135)(133,134);(104,105)(106,109)(107,114)(108,111)(110,116)(112,119)(113,118)(115,117)(120,121)(122,128)(123,127)(124,126)(125,129)(130,134)(131,133)(132,135)Polycyclic presentation:
2^7,103|b^2=g,c^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dfg,c^b=cfg,d^b=df,h^d=h^1021466 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13184, 18655);