Finite Group DatabasebetaFinite Group Databasebeta| order | 21504 |
| isoclinism | isoclinic class of #72209888 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C27 ⋊ GL(3,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #72209662 |
| commutator, hypocenter | G | C1 |
| Frattini, socle | C24 | AGL(3,𝔽2) |
| Fitting, cosocle | C27 | GL(3,𝔽2) |
| LCS | G | |
| UCS | C1 ◁ C2 | |
2126 conjugacy classes of subgroups; only the 6 normal subgroups are shown.
This group has 2126 conjugacy classes of subgroups — too many to draw. Only its 6 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Permutation generators:
(1,2)(3,8)(4,13)(5,9)(7,18)(10,14)(11,26)(12,24)(15,17)(16,30)(19,36)(20,34)(21,39)(23,40)(25,27)(29,37)(35,38)(41,52)(43,53)(45,46)(50,55)(51,54)(58,62)(59,66)(60,65)(63,71)(67,70)(68,72);(2,6,7)(3,11,12)(4,14,5)(8,22,23)(9,17,20)(10,15,21)(13,28,29)(16,19,25)(18,32,33)(24,30,41)(26,31,38)(34,43,49)(35,42,50)(36,51,45)(37,53,48)(39,44,47)(40,54,55)(46,52,56)(58,59,63)(64,67,66)(65,68,71)(69,70,72);(2,8)(6,15)(7,19)(9,24)(10,25)(13,26)(16,22)(17,28)(18,34)(20,38)(21,23)(29,41)(30,31)(32,42)(33,45)(35,36)(37,39)(40,52)(43,51)(44,56)(46,48)(47,55)(49,50)(53,54)(57,58)(59,63)(60,64)(61,65)(62,69)(66,67)(68,71)(70,72);(2,8)(6,16)(9,24)(10,25)(13,26)(15,22)(17,31)(18,35)(28,30)(33,45)(34,36)(37,40)(39,52)(46,48)(47,55)(49,50)(57,59)(58,63)(60,67)(61,68)(62,70)(64,66)(65,71)(69,72);(1,3)(4,11)(5,12)(6,17)(7,20)(14,27)(15,28)(16,31)(18,36)(19,38)(21,29)(22,30)(23,41)(32,42)(33,46)(34,35)(37,52)(39,40)(43,51)(44,56)(45,48)(47,50)(49,55)(53,54)(57,60)(58,64)(59,67)(61,62)(63,66)(65,69)(68,70)(71,72);(57,61)(58,65)(59,68)(60,62)(63,71)(64,69)(66,72)(67,70);(1,4)(2,9)(3,11)(5,14)(6,17)(8,24)(10,13)(12,27)(15,28)(16,31)(18,37)(22,30)(25,26)(32,43)(33,47)(34,39)(35,40)(36,52)(42,51)(44,53)(45,55)(46,50)(48,49)(54,56);(1,5)(2,9)(3,12)(4,14)(7,20)(8,24)(10,13)(11,27)(18,34)(19,38)(21,29)(23,41)(25,26)(32,44)(33,48)(35,36)(37,39)(40,52)(42,56)(43,53)(45,46)(47,49)(50,55)(51,54);(2,10)(6,15)(7,21)(8,25)(9,13)(16,22)(17,28)(18,34)(19,23)(20,29)(24,26)(30,31)(32,43)(33,49)(35,36)(37,39)(38,41)(40,52)(42,51)(44,53)(45,50)(46,55)(47,48)(54,56)32 conjugacy classes — view the full character table.
| maximal quotient of | #72227918, #72228028 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:
gap> G := Group((1,2)(3,8)(4,13)(5,9)(7,18)(10,14)(11,26)(12,24)(15,17)(16,30)(19,36)(20,34)(21,39)(23,40)(25,27)(29,37)(35,38)(41,52)(43,53)(45,46)(50,55)(51,54)(58,62)(59,66)(60,65)(63,71)(67,70)(68,72), (2,6,7)(3,11,12)(4,14,5)(8,22,23)(9,17,20)(10,15,21)(13,28,29)(16,19,25)(18,32,33)(24,30,41)(26,31,38)(34,43,49)(35,42,50)(36,51,45)(37,53,48)(39,44,47)(40,54,55)(46,52,56)(58,59,63)(64,67,66)(65,68,71)(69,70,72), (2,8)(6,15)(7,19)(9,24)(10,25)(13,26)(16,22)(17,28)(18,34)(20,38)(21,23)(29,41)(30,31)(32,42)(33,45)(35,36)(37,39)(40,52)(43,51)(44,56)(46,48)(47,55)(49,50)(53,54)(57,58)(59,63)(60,64)(61,65)(62,69)(66,67)(68,71)(70,72), (2,8)(6,16)(9,24)(10,25)(13,26)(15,22)(17,31)(18,35)(28,30)(33,45)(34,36)(37,40)(39,52)(46,48)(47,55)(49,50)(57,59)(58,63)(60,67)(61,68)(62,70)(64,66)(65,71)(69,72), (1,3)(4,11)(5,12)(6,17)(7,20)(14,27)(15,28)(16,31)(18,36)(19,38)(21,29)(22,30)(23,41)(32,42)(33,46)(34,35)(37,52)(39,40)(43,51)(44,56)(45,48)(47,50)(49,55)(53,54)(57,60)(58,64)(59,67)(61,62)(63,66)(65,69)(68,70)(71,72), (57,61)(58,65)(59,68)(60,62)(63,71)(64,69)(66,72)(67,70), (1,4)(2,9)(3,11)(5,14)(6,17)(8,24)(10,13)(12,27)(15,28)(16,31)(18,37)(22,30)(25,26)(32,43)(33,47)(34,39)(35,40)(36,52)(42,51)(44,53)(45,55)(46,50)(48,49)(54,56), (1,5)(2,9)(3,12)(4,14)(7,20)(8,24)(10,13)(11,27)(18,34)(19,38)(21,29)(23,41)(25,26)(32,44)(33,48)(35,36)(37,39)(40,52)(42,56)(43,53)(45,46)(47,49)(50,55)(51,54), (2,10)(6,15)(7,21)(8,25)(9,13)(16,22)(17,28)(18,34)(19,23)(20,29)(24,26)(30,31)(32,43)(33,49)(35,36)(37,39)(38,41)(40,52)(42,51)(44,53)(45,50)(46,55)(47,48)(54,56));