Finite Group DatabasebetaFinite Group Databasebeta| order | 7680 |
| isoclinism | isoclinic class of #72209628 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C4 | #11757162 |
| commutator, hypocenter | G | C1 |
| Fitting, cosocle | C2×#585 | A5 |
| socle | C22 | #14515300 |
| LCS | G | |
| UCS | C1 ◁ C2×C4 | |
505 conjugacy classes of subgroups; only the 10 normal subgroups are shown.
This group has 505 conjugacy classes of subgroups — too many to draw. Only its 10 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Permutation generators:
(1,2,5,3)(4,7,6,8)(9,13,11,14)(10,15,12,16)(17,19,18,20)(21,24,23,22)(25,26,32,27)(28,35,44,39)(29,36,47,40)(30,33,52,37)(31,34,56,38)(41,63,51,54)(42,59,75,68)(43,62,76,49)(45,57,79,66)(46,65,55,60)(48,81,82,53)(50,58,70,67)(61,71,69,64)(72,86,84,74)(73,88,78,83)(77,80,87,85);(1,4,2)(3,5,6)(7,9,10)(8,11,12)(13,16,17)(14,15,18)(19,21,22)(20,23,24)(25,28,29)(26,33,34)(27,37,38)(30,48,49)(31,53,54)(32,44,47)(35,61,50)(36,64,51)(39,69,70)(40,71,41)(42,72,74)(45,65,77)(46,80,59)(52,82,62)(55,85,68)(56,81,63)(57,83,78)(60,87,79)(66,88,73)(75,84,86);(1,5)(7,8)(9,11)(10,12)(13,14)(15,16)(17,18)(22,24)(25,30)(26,35)(27,39)(28,41)(29,45)(31,50)(32,52)(33,54)(34,59)(36,62)(37,63)(38,68)(40,49)(42,72)(43,73)(44,51)(46,53)(47,79)(48,77)(55,81)(56,70)(57,83)(58,74)(60,64)(61,80)(65,71)(66,88)(67,86)(69,85)(75,84)(76,78)(82,87);(2,3)(4,6)(7,8)(9,11)(13,14)(17,18)(19,20)(21,23)(25,31)(26,36)(27,40)(28,42)(29,46)(30,50)(32,56)(33,57)(34,60)(35,62)(37,66)(38,65)(39,49)(41,72)(43,61)(44,75)(45,53)(47,55)(48,67)(51,84)(52,70)(54,83)(58,82)(59,64)(63,88)(68,71)(69,76)(73,80)(74,87)(77,86)(78,85)(79,81);(2,3)(4,6)(9,11)(10,12)(13,14)(15,16)(19,20)(21,23)(25,28)(26,33)(27,37)(29,43)(30,51)(31,42)(32,44)(34,58)(35,63)(36,57)(38,67)(39,54)(40,66)(41,52)(45,78)(46,61)(47,76)(48,65)(49,83)(50,84)(53,85)(55,69)(56,75)(59,86)(60,82)(62,88)(64,77)(68,74)(70,72)(71,87)(73,79)(80,81);(1,5)(4,6)(7,8)(9,11)(13,14)(17,18)(19,20)(22,24)(25,29)(26,34)(27,38)(28,43)(30,45)(31,55)(32,47)(33,58)(35,59)(36,65)(37,67)(39,68)(40,60)(41,73)(42,69)(44,76)(46,56)(48,57)(49,64)(50,81)(51,78)(52,79)(53,70)(54,74)(61,75)(62,71)(63,86)(66,82)(72,85)(77,83)(80,84)(87,88);(1,5)(2,3)(4,6)(7,8)(9,11)(10,12)(13,14)(15,16)(17,18)(19,20)(21,23)(22,24)(25,32)(26,27)(28,44)(29,47)(30,52)(31,56)(33,37)(34,38)(35,39)(36,40)(41,51)(42,75)(43,76)(45,79)(46,55)(48,82)(49,62)(50,70)(53,81)(54,63)(57,66)(58,67)(59,68)(60,65)(61,69)(64,71)(72,84)(73,78)(74,86)(77,87)(80,85)(83,88);(25,26,32,27)(28,33,44,37)(29,34,47,38)(30,35,52,39)(31,36,56,40)(41,54,51,63)(42,57,75,66)(43,58,76,67)(45,59,79,68)(46,60,55,65)(48,61,82,69)(49,50,62,70)(53,64,81,71)(72,83,84,88)(73,74,78,86)(77,80,87,85)50 conjugacy classes — view the full character table.
| maximal quotient of | #72209698 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:
gap> G := Group((1,2,5,3)(4,7,6,8)(9,13,11,14)(10,15,12,16)(17,19,18,20)(21,24,23,22)(25,26,32,27)(28,35,44,39)(29,36,47,40)(30,33,52,37)(31,34,56,38)(41,63,51,54)(42,59,75,68)(43,62,76,49)(45,57,79,66)(46,65,55,60)(48,81,82,53)(50,58,70,67)(61,71,69,64)(72,86,84,74)(73,88,78,83)(77,80,87,85), (1,4,2)(3,5,6)(7,9,10)(8,11,12)(13,16,17)(14,15,18)(19,21,22)(20,23,24)(25,28,29)(26,33,34)(27,37,38)(30,48,49)(31,53,54)(32,44,47)(35,61,50)(36,64,51)(39,69,70)(40,71,41)(42,72,74)(45,65,77)(46,80,59)(52,82,62)(55,85,68)(56,81,63)(57,83,78)(60,87,79)(66,88,73)(75,84,86), (1,5)(7,8)(9,11)(10,12)(13,14)(15,16)(17,18)(22,24)(25,30)(26,35)(27,39)(28,41)(29,45)(31,50)(32,52)(33,54)(34,59)(36,62)(37,63)(38,68)(40,49)(42,72)(43,73)(44,51)(46,53)(47,79)(48,77)(55,81)(56,70)(57,83)(58,74)(60,64)(61,80)(65,71)(66,88)(67,86)(69,85)(75,84)(76,78)(82,87), (2,3)(4,6)(7,8)(9,11)(13,14)(17,18)(19,20)(21,23)(25,31)(26,36)(27,40)(28,42)(29,46)(30,50)(32,56)(33,57)(34,60)(35,62)(37,66)(38,65)(39,49)(41,72)(43,61)(44,75)(45,53)(47,55)(48,67)(51,84)(52,70)(54,83)(58,82)(59,64)(63,88)(68,71)(69,76)(73,80)(74,87)(77,86)(78,85)(79,81), (2,3)(4,6)(9,11)(10,12)(13,14)(15,16)(19,20)(21,23)(25,28)(26,33)(27,37)(29,43)(30,51)(31,42)(32,44)(34,58)(35,63)(36,57)(38,67)(39,54)(40,66)(41,52)(45,78)(46,61)(47,76)(48,65)(49,83)(50,84)(53,85)(55,69)(56,75)(59,86)(60,82)(62,88)(64,77)(68,74)(70,72)(71,87)(73,79)(80,81), (1,5)(4,6)(7,8)(9,11)(13,14)(17,18)(19,20)(22,24)(25,29)(26,34)(27,38)(28,43)(30,45)(31,55)(32,47)(33,58)(35,59)(36,65)(37,67)(39,68)(40,60)(41,73)(42,69)(44,76)(46,56)(48,57)(49,64)(50,81)(51,78)(52,79)(53,70)(54,74)(61,75)(62,71)(63,86)(66,82)(72,85)(77,83)(80,84)(87,88), (1,5)(2,3)(4,6)(7,8)(9,11)(10,12)(13,14)(15,16)(17,18)(19,20)(21,23)(22,24)(25,32)(26,27)(28,44)(29,47)(30,52)(31,56)(33,37)(34,38)(35,39)(36,40)(41,51)(42,75)(43,76)(45,79)(46,55)(48,82)(49,62)(50,70)(53,81)(54,63)(57,66)(58,67)(59,68)(60,65)(61,69)(64,71)(72,84)(73,78)(74,86)(77,87)(80,85)(83,88), (25,26,32,27)(28,33,44,37)(29,34,47,38)(30,35,52,39)(31,36,56,40)(41,54,51,63)(42,57,75,66)(43,58,76,67)(45,59,79,68)(46,60,55,65)(48,61,82,69)(49,50,62,70)(53,64,81,71)(72,83,84,88)(73,74,78,86)(77,80,87,85));