Finite Group DatabasebetaFinite Group Databasebeta| order | 8748 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C34 |
| direct product | C2×#72221890 |
| Fitting | C2×#14554343 ⋊ C2 |
| Sylow | #14554343 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C3×#92301 |
| commutator, cosocle, hypocenter | H(5,𝔽3) | C22×C32 |
| Frattini | C33 | C2×C32×C3⋊D3 |
| Fitting | C2×#14554343 | C2 |
| hypercenter | C2×C32 | #92301 |
| LCS | G ▷ H(5,𝔽3) | |
| UCS | C1 ◁ C2×C3 ◁ C2×C32 | |
3744 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 3744 conjugacy classes of subgroups — too many to draw. Only its 160 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 83 |
| transitive degree | μt(G) = 162 |
A minimal faithful permutation representation — 3 generators on 83 points:
(1,7,23,5,11,8,6,2,21)(3,26,24,46,22,59,15,32,27,12,9,48,20,47,35,34,13,51)(4,57,30,45,10,71,19,41,25,16,31,56,18,29,43,42,17,50)(14,54,49,66,44,81,40,64,53,37,28,74,39,72,58,67,36,78)(33,65,77,70,52,79,62,68,73,38,60,80,63,55,76,69,61,75);(1,4,16)(2,17,57)(3,14,38)(5,18,45)(6,19,42)(7,31,29)(8,43,56)(9,44,65)(10,41,11)(12,33,37)(13,28,68)(15,40,69)(20,39,70)(21,25,71)(22,36,55)(23,30,50)(24,53,80)(26,61,64)(27,58,75)(32,52,72)(34,62,67)(35,49,79)(46,63,66)(47,60,54)(48,76,81)(51,77,74)(59,73,78)(82,83);(1,3,12)(2,9,26)(4,14,33)(5,20,46)(6,15,34)(7,22,47)(8,24,48)(10,28,52)(11,13,32)(16,38,37)(17,44,61)(18,39,63)(19,40,62)(21,35,59)(23,27,51)(25,49,73)(29,55,54)(30,58,77)(31,36,60)(41,68,72)(42,69,67)(43,53,76)(45,70,66)(50,75,74)(56,80,81)(57,65,64)(71,79,78)188 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72221899 |
| maximal normal subgroup of | C2×#72221899 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,7,23,5,11,8,6,2,21)(3,26,24,46,22,59,15,32,27,12,9,48,20,47,35,34,13,51)(4,57,30,45,10,71,19,41,25,16,31,56,18,29,43,42,17,50)(14,54,49,66,44,81,40,64,53,37,28,74,39,72,58,67,36,78)(33,65,77,70,52,79,62,68,73,38,60,80,63,55,76,69,61,75), (1,4,16)(2,17,57)(3,14,38)(5,18,45)(6,19,42)(7,31,29)(8,43,56)(9,44,65)(10,41,11)(12,33,37)(13,28,68)(15,40,69)(20,39,70)(21,25,71)(22,36,55)(23,30,50)(24,53,80)(26,61,64)(27,58,75)(32,52,72)(34,62,67)(35,49,79)(46,63,66)(47,60,54)(48,76,81)(51,77,74)(59,73,78)(82,83), (1,3,12)(2,9,26)(4,14,33)(5,20,46)(6,15,34)(7,22,47)(8,24,48)(10,28,52)(11,13,32)(16,38,37)(17,44,61)(18,39,63)(19,40,62)(21,35,59)(23,27,51)(25,49,73)(29,55,54)(30,58,77)(31,36,60)(41,68,72)(42,69,67)(43,53,76)(45,70,66)(50,75,74)(56,80,81)(57,65,64)(71,79,78));