Finite Group DatabasebetaFinite Group Databasebeta| order | 7680 |
| isoclinism | isoclinic class of #72209630 |
| 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 | |
485 conjugacy classes of subgroups; only the 11 normal subgroups are shown.
This group has 485 conjugacy classes of subgroups — too many to draw. Only its 11 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)(26,28)(27,29)(32,53)(33,55)(34,45)(35,46)(36,41)(37,47)(38,48)(40,68)(42,49)(43,50)(44,51)(54,81)(56,77)(58,74)(59,76)(60,71)(61,67)(62,72)(63,80)(64,69)(65,70)(66,75)(78,79)(82,88)(83,86)(85,87);(1,4,2)(3,5,6)(7,9,10)(8,11,12)(13,16,17)(14,15,18)(19,21,22)(20,23,24)(26,32,33)(27,39,40)(28,45,29)(34,61,41)(35,64,58)(36,54,68)(37,49,59)(38,42,60)(43,57,66)(44,73,72)(46,77,63)(47,74,50)(48,80,51)(53,81,70)(55,82,83)(56,76,62)(65,87,78)(67,86,85)(69,71,75)(79,84,88);(1,5)(2,3)(4,6)(7,8)(9,11)(10,12)(13,14)(15,16)(17,18)(19,20)(21,23)(22,24);(25,26)(27,34)(28,46)(29,36)(30,37)(31,38)(32,33)(35,65)(39,71)(40,74)(41,75)(42,60)(43,58)(44,63)(45,72)(47,61)(48,67)(49,59)(50,69)(51,56)(52,70)(53,81)(54,73)(55,57)(62,86)(64,82)(66,87)(68,77)(76,84)(78,83)(79,85)(80,88);(25,27)(26,34)(28,41)(29,49)(30,43)(31,44)(32,51)(33,56)(35,62)(36,59)(37,58)(38,63)(39,72)(40,57)(42,78)(45,71)(46,75)(47,64)(48,77)(50,53)(52,79)(54,80)(55,74)(60,83)(61,82)(65,86)(66,84)(67,68)(69,81)(70,85)(73,88)(76,87);(25,28)(26,35)(27,41)(29,45)(30,47)(31,48)(32,54)(33,57)(34,62)(36,66)(37,67)(38,61)(39,60)(40,56)(42,76)(43,64)(44,77)(46,70)(49,71)(50,74)(51,80)(52,65)(53,55)(58,68)(59,84)(63,82)(69,88)(72,83)(73,81)(75,85)(78,87)(79,86);(25,29)(26,36)(27,42)(28,45)(30,50)(31,51)(32,43)(33,58)(34,60)(35,66)(37,69)(38,56)(39,62)(40,61)(41,76)(44,53)(46,72)(47,74)(48,80)(49,79)(52,78)(54,64)(55,77)(57,68)(59,85)(63,81)(65,87)(67,88)(70,83)(71,86)(73,82)(75,84);(25,30,52,31)(26,37,70,38)(27,43,79,44)(28,47,65,48)(29,50,78,51)(32,49,53,42)(33,59,81,60)(34,58,85,63)(35,67,46,61)(36,69,83,56)(39,57,84,73)(40,66,88,72)(41,64,86,77)(45,74,87,80)(54,71,55,76)(62,68,75,82)46 conjugacy classes — view the full character table.
| maximal subgroup of | #72227900 |
| 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)(26,28)(27,29)(32,53)(33,55)(34,45)(35,46)(36,41)(37,47)(38,48)(40,68)(42,49)(43,50)(44,51)(54,81)(56,77)(58,74)(59,76)(60,71)(61,67)(62,72)(63,80)(64,69)(65,70)(66,75)(78,79)(82,88)(83,86)(85,87), (1,4,2)(3,5,6)(7,9,10)(8,11,12)(13,16,17)(14,15,18)(19,21,22)(20,23,24)(26,32,33)(27,39,40)(28,45,29)(34,61,41)(35,64,58)(36,54,68)(37,49,59)(38,42,60)(43,57,66)(44,73,72)(46,77,63)(47,74,50)(48,80,51)(53,81,70)(55,82,83)(56,76,62)(65,87,78)(67,86,85)(69,71,75)(79,84,88), (1,5)(2,3)(4,6)(7,8)(9,11)(10,12)(13,14)(15,16)(17,18)(19,20)(21,23)(22,24), (25,26)(27,34)(28,46)(29,36)(30,37)(31,38)(32,33)(35,65)(39,71)(40,74)(41,75)(42,60)(43,58)(44,63)(45,72)(47,61)(48,67)(49,59)(50,69)(51,56)(52,70)(53,81)(54,73)(55,57)(62,86)(64,82)(66,87)(68,77)(76,84)(78,83)(79,85)(80,88), (25,27)(26,34)(28,41)(29,49)(30,43)(31,44)(32,51)(33,56)(35,62)(36,59)(37,58)(38,63)(39,72)(40,57)(42,78)(45,71)(46,75)(47,64)(48,77)(50,53)(52,79)(54,80)(55,74)(60,83)(61,82)(65,86)(66,84)(67,68)(69,81)(70,85)(73,88)(76,87), (25,28)(26,35)(27,41)(29,45)(30,47)(31,48)(32,54)(33,57)(34,62)(36,66)(37,67)(38,61)(39,60)(40,56)(42,76)(43,64)(44,77)(46,70)(49,71)(50,74)(51,80)(52,65)(53,55)(58,68)(59,84)(63,82)(69,88)(72,83)(73,81)(75,85)(78,87)(79,86), (25,29)(26,36)(27,42)(28,45)(30,50)(31,51)(32,43)(33,58)(34,60)(35,66)(37,69)(38,56)(39,62)(40,61)(41,76)(44,53)(46,72)(47,74)(48,80)(49,79)(52,78)(54,64)(55,77)(57,68)(59,85)(63,81)(65,87)(67,88)(70,83)(71,86)(73,82)(75,84), (25,30,52,31)(26,37,70,38)(27,43,79,44)(28,47,65,48)(29,50,78,51)(32,49,53,42)(33,59,81,60)(34,58,85,63)(35,67,46,61)(36,69,83,56)(39,57,84,73)(40,66,88,72)(41,64,86,77)(45,74,87,80)(54,71,55,76)(62,68,75,82));