Finite Group DatabasebetaFinite Group Databasebeta| order | 23226 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (C49⋊C3)⋊D79 |
| Fitting, derived, Hall | C49×C79 ⋊ C2×C3 |
| Hall | C79×C49⋊C3 ⋊ C2 |
| Hall | D3871 ⋊ C3 |
| Hall | C49⋊C3 ⋊ D79 |
| Sylow | C79 ⋊ D49⋊C3 |
| Sylow | C49 ⋊ C3×D79 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C49×C79 | C2×C3 |
| Frattini | C7 | ΓL(1,𝔽8)⋊D79 |
| socle | C7×C79 | AGL(1,𝔽7) |
| LCS | G ▷ C49×C79 | |
| UCS | C1 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 3871 |
A minimal faithful permutation representation — 2 generators on 128 points:
(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)(81,89,91,83,84,92)(82,101,103,87,88,104)(85,113,100,94,95,122)(86,118,119,98,99,128)(90,97,96,102,107,115)(93,116,125,111,112,126)(105,117,120,123,106,109)(108,121,110,124,114,127);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(80,102,99,110,109,114,83,87,124,93,92,119,94,98,101,125,84,113,115,107,122,103,123,117,116,112,106,105,104,100,97,96,95,89,126,88,86,85,128,91,111,108,82,81,121,120,127,118,90)Polycyclic presentation:
2,3,7,79,7|c^7=e^6,c^a=c^6e,d^a=d^78,e^a=e^6,c^b=c^2e^3,e^b=e^2755 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(23226, 22);