Finite Group DatabasebetaFinite Group Databasebeta| order | 61440 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | #6342371 ⋊ S5 |
| derived | #72209721 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | #72209721 | C22 |
| Frattini, socle | C24 | C2×#11757163⋊C2 |
| Fitting | #6342371 | S5 |
| LCS | G ▷ #72209721 | |
| UCS | C1 | |
At least 26552 conjugacy classes of subgroups; only the 10 normal subgroups are shown.
This group has at least 26552 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:
(1,7)(9,32)(10,33)(12,35)(14,37)(15,38)(18,40)(20,42)(22,44)(24,46)(26,47)(28,61)(30,49)(51,53)(55,56)(57,63);(2,8)(9,32)(10,33)(11,34)(12,35)(13,36)(15,38)(16,39)(21,43)(22,44)(27,60)(28,61)(50,52)(51,53)(57,63)(59,62);(3,17)(10,33)(11,34)(12,35)(13,36)(18,40)(19,41)(20,42)(27,60)(28,61)(29,48)(30,49)(50,52)(51,53)(54,58)(55,56);(4,25)(12,35)(13,36)(15,38)(16,39)(19,41)(20,42)(23,45)(24,46)(26,47)(29,48)(30,49)(50,52)(51,53)(57,63)(59,62);(5,31)(14,37)(15,38)(16,39)(21,43)(22,44)(23,45)(24,46)(27,60)(28,61)(29,48)(30,49)(50,52)(51,53)(54,58)(55,56);(2,9)(5,14)(8,32)(10,11)(12,13)(23,24)(29,30)(31,37)(33,34)(35,36)(45,46)(48,49)(54,55)(56,58)(57,59)(62,63);(2,9)(5,14)(8,32)(10,11)(12,13)(23,24)(29,30)(31,37)(33,34)(35,36)(45,46)(48,49)(54,55)(56,58)(57,59)(62,63);(3,18)(5,14)(10,11)(12,13)(15,16)(17,40)(19,20)(21,22)(23,24)(31,37)(33,34)(35,36)(38,39)(41,42)(43,44)(45,46);(2,10,8,33)(4,19,25,41)(5,14)(9,11,32,34)(12,36)(13,35)(15,38)(16,39)(20,47,42,26)(21,27,43,60)(22,28,44,61)(23,30,45,49)(24,29,46,48)(31,37)(50,52)(51,53)(54,55)(56,58)(57,62)(59,63);(2,11,32,33)(3,18)(4,19,47,42)(5,14)(8,34,9,10)(12,36)(13,35)(15,39)(16,38)(17,40)(20,25,41,26)(21,28,44,60)(22,27,43,61)(23,30,46,48)(24,29,45,49)(31,37)(50,52)(51,53)(57,63)(59,62);(2,12,34,9,13,33)(3,19,4)(5,14)(8,35,11,32,36,10)(17,41,25)(18,20,26)(21,50,60)(22,51,61)(23,56,29,24,58,30)(27,43,52)(28,44,53)(31,37)(40,42,47)(45,55,48,46,54,49)(57,59)(62,63);(2,13,33)(3,20,4)(5,14)(8,36,10)(9,12,34)(11,32,35)(15,16)(17,42,25)(18,19,26)(21,51,61,22,50,60)(23,56,30,24,58,29)(27,43,53,28,44,52)(31,37)(38,39)(40,41,47)(45,55,49,46,54,48);(2,14,13,33)(3,21,52,4)(5,12,34,9)(8,37,36,10)(11,32,31,35)(15,38)(16,39)(17,43,50,25)(18,22,53,26)(19,27,41,60)(20,28,42,61)(23,57,58,49)(24,59,56,48)(29,46,62,55)(30,45,63,54)(40,44,51,47);(2,5,12,34,9,14,13,33)(3,22,52,4)(8,31,35,11,32,37,36,10)(15,39)(16,38)(17,44,50,25)(18,21,53,26)(19,28,42,60)(20,27,41,61)(23,57,56,49,24,59,58,48)(29,45,63,55,30,46,62,54)(40,43,51,47);(2,15,14,13,33)(3,23,57,52,4)(5,12,34,9,16)(8,38,37,36,10)(11,32,39,31,35)(17,45,63,50,25)(18,24,59,53,26)(19,48,44,56,60)(20,49,43,58,61)(21,54,28,42,30)(22,55,27,41,29)(40,46,62,51,47);(2,16,14,13,33)(3,24,59,52,4)(5,12,34,9,15)(8,39,37,36,10)(11,32,38,31,35)(17,46,62,50,25)(18,23,57,53,26)(19,49,44,58,60)(20,48,43,56,61)(21,55,28,42,29)(22,54,27,41,30)(40,45,63,51,47);(3,18)(5,14)(10,11)(12,13)(15,16)(17,40)(19,20)(21,22)(23,24)(31,37)(33,34)(35,36)(38,39)(41,42)(43,44)(45,46);(4,26)(5,14)(12,13)(19,20)(21,22)(25,47)(27,28)(31,37)(35,36)(41,42)(43,44)(54,55)(56,58)(57,59)(60,61)(62,63);(3,20,17,42)(5,14)(10,13,33,36)(11,12,34,35)(15,16)(18,19,40,41)(21,22)(23,24)(27,50,60,52)(28,51,61,53)(29,54,48,58)(30,55,49,56)(31,37)(38,39)(43,44)(45,46);(3,19,40,42)(4,26)(5,14)(10,12,34,36)(11,13,33,35)(17,41,18,20)(21,22)(25,47)(27,51,61,52)(28,50,60,53)(29,54,49,56)(30,55,48,58)(31,37)(43,44)(57,59)(62,63);(3,22,19,40,43,42)(4,27,52)(5,12,34,37,36,10)(11,14,13,33,31,35)(15,39)(16,38)(17,44,41,18,21,20)(23,46)(24,45)(25,60,50)(26,28,53)(29,62,54)(30,63,55)(47,61,51)(48,59,58)(49,57,56);(3,21,20,17,43,42)(4,28,52)(5,12,34,31,35,11)(10,14,13,33,37,36)(15,38)(16,39)(18,22,19,40,44,41)(23,45)(24,46)(25,61,50)(26,27,53)(29,62,55)(30,63,54)(47,60,51)(48,59,56)(49,57,58);(3,24,43,42,17,46,21,20)(4,29,59,50)(5,12,11,15,31,35,34,38)(10,16,37,36,33,39,14,13)(18,23,44,41,40,45,22,19)(25,48,62,52)(26,30,57,51)(27,54)(28,55)(47,49,63,53)(56,61)(58,60);(3,23,43,42,17,45,21,20)(4,30,57,50)(5,12,11,16,31,35,34,39)(10,15,37,36,33,38,14,13)(18,24,44,41,40,46,22,19)(25,49,63,52)(26,29,59,51)(27,55)(28,54)(47,48,62,53)(56,60)(58,61);(4,26)(5,14)(12,13)(19,20)(21,22)(25,47)(27,28)(31,37)(35,36)(41,42)(43,44)(54,55)(56,58)(57,59)(60,61)(62,63);(5,14)(15,16)(21,22)(23,24)(27,28)(29,30)(31,37)(38,39)(43,44)(45,46)(48,49)(50,51)(52,53)(54,55)(56,58)(60,61);(4,28)(5,12)(13,14)(15,38)(16,39)(19,22)(20,21)(23,45)(24,46)(25,61)(26,27)(29,48)(30,49)(31,35)(36,37)(41,44)(42,43)(47,60)(50,52)(51,53)(54,63)(55,62)(56,59)(57,58);(4,27,26,28)(5,12,14,13)(15,39)(16,38)(19,21,20,22)(23,46)(24,45)(25,60,47,61)(29,49)(30,48)(31,35,37,36)(41,43,42,44)(50,53)(51,52)(54,63,55,62)(56,59,58,57);(4,30,28)(5,12,39)(13,38,14)(15,37,36)(16,31,35)(19,46,22)(20,45,21)(23,43,42)(24,44,41)(25,49,61)(26,29,27)(47,48,60)(50,58,57)(51,56,59)(52,54,63)(53,55,62);(4,29,28)(5,12,38)(13,39,14)(15,31,35)(16,37,36)(19,45,22)(20,46,21)(23,44,41)(24,43,42)(25,48,61)(26,30,27)(47,49,60)(50,56,59)(51,58,57)(52,55,62)(53,54,63);(5,14)(15,16)(21,22)(23,24)(27,28)(29,30)(31,37)(38,39)(43,44)(45,46)(48,49)(50,51)(52,53)(54,55)(56,58)(60,61);(5,15,31,38)(14,16,37,39)(21,24,43,46)(22,23,44,45)(27,49,60,30)(28,48,61,29)(50,56,52,55)(51,58,53,54);(5,16,31,39)(14,15,37,38)(21,23,43,45)(22,24,44,46)(27,48,60,29)(28,49,61,30)(50,58,52,54)(51,56,53,55)69 conjugacy classes — view the full character table.
| automorphism group of | C2×2−1+4 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:
gap> G := Group((1,7)(9,32)(10,33)(12,35)(14,37)(15,38)(18,40)(20,42)(22,44)(24,46)(26,47)(28,61)(30,49)(51,53)(55,56)(57,63), (2,8)(9,32)(10,33)(11,34)(12,35)(13,36)(15,38)(16,39)(21,43)(22,44)(27,60)(28,61)(50,52)(51,53)(57,63)(59,62), (3,17)(10,33)(11,34)(12,35)(13,36)(18,40)(19,41)(20,42)(27,60)(28,61)(29,48)(30,49)(50,52)(51,53)(54,58)(55,56), (4,25)(12,35)(13,36)(15,38)(16,39)(19,41)(20,42)(23,45)(24,46)(26,47)(29,48)(30,49)(50,52)(51,53)(57,63)(59,62), (5,31)(14,37)(15,38)(16,39)(21,43)(22,44)(23,45)(24,46)(27,60)(28,61)(29,48)(30,49)(50,52)(51,53)(54,58)(55,56), (2,9)(5,14)(8,32)(10,11)(12,13)(23,24)(29,30)(31,37)(33,34)(35,36)(45,46)(48,49)(54,55)(56,58)(57,59)(62,63), (2,9)(5,14)(8,32)(10,11)(12,13)(23,24)(29,30)(31,37)(33,34)(35,36)(45,46)(48,49)(54,55)(56,58)(57,59)(62,63), (3,18)(5,14)(10,11)(12,13)(15,16)(17,40)(19,20)(21,22)(23,24)(31,37)(33,34)(35,36)(38,39)(41,42)(43,44)(45,46), (2,10,8,33)(4,19,25,41)(5,14)(9,11,32,34)(12,36)(13,35)(15,38)(16,39)(20,47,42,26)(21,27,43,60)(22,28,44,61)(23,30,45,49)(24,29,46,48)(31,37)(50,52)(51,53)(54,55)(56,58)(57,62)(59,63), (2,11,32,33)(3,18)(4,19,47,42)(5,14)(8,34,9,10)(12,36)(13,35)(15,39)(16,38)(17,40)(20,25,41,26)(21,28,44,60)(22,27,43,61)(23,30,46,48)(24,29,45,49)(31,37)(50,52)(51,53)(57,63)(59,62), (2,12,34,9,13,33)(3,19,4)(5,14)(8,35,11,32,36,10)(17,41,25)(18,20,26)(21,50,60)(22,51,61)(23,56,29,24,58,30)(27,43,52)(28,44,53)(31,37)(40,42,47)(45,55,48,46,54,49)(57,59)(62,63), (2,13,33)(3,20,4)(5,14)(8,36,10)(9,12,34)(11,32,35)(15,16)(17,42,25)(18,19,26)(21,51,61,22,50,60)(23,56,30,24,58,29)(27,43,53,28,44,52)(31,37)(38,39)(40,41,47)(45,55,49,46,54,48), (2,14,13,33)(3,21,52,4)(5,12,34,9)(8,37,36,10)(11,32,31,35)(15,38)(16,39)(17,43,50,25)(18,22,53,26)(19,27,41,60)(20,28,42,61)(23,57,58,49)(24,59,56,48)(29,46,62,55)(30,45,63,54)(40,44,51,47), (2,5,12,34,9,14,13,33)(3,22,52,4)(8,31,35,11,32,37,36,10)(15,39)(16,38)(17,44,50,25)(18,21,53,26)(19,28,42,60)(20,27,41,61)(23,57,56,49,24,59,58,48)(29,45,63,55,30,46,62,54)(40,43,51,47), (2,15,14,13,33)(3,23,57,52,4)(5,12,34,9,16)(8,38,37,36,10)(11,32,39,31,35)(17,45,63,50,25)(18,24,59,53,26)(19,48,44,56,60)(20,49,43,58,61)(21,54,28,42,30)(22,55,27,41,29)(40,46,62,51,47), (2,16,14,13,33)(3,24,59,52,4)(5,12,34,9,15)(8,39,37,36,10)(11,32,38,31,35)(17,46,62,50,25)(18,23,57,53,26)(19,49,44,58,60)(20,48,43,56,61)(21,55,28,42,29)(22,54,27,41,30)(40,45,63,51,47), (3,18)(5,14)(10,11)(12,13)(15,16)(17,40)(19,20)(21,22)(23,24)(31,37)(33,34)(35,36)(38,39)(41,42)(43,44)(45,46), (4,26)(5,14)(12,13)(19,20)(21,22)(25,47)(27,28)(31,37)(35,36)(41,42)(43,44)(54,55)(56,58)(57,59)(60,61)(62,63), (3,20,17,42)(5,14)(10,13,33,36)(11,12,34,35)(15,16)(18,19,40,41)(21,22)(23,24)(27,50,60,52)(28,51,61,53)(29,54,48,58)(30,55,49,56)(31,37)(38,39)(43,44)(45,46), (3,19,40,42)(4,26)(5,14)(10,12,34,36)(11,13,33,35)(17,41,18,20)(21,22)(25,47)(27,51,61,52)(28,50,60,53)(29,54,49,56)(30,55,48,58)(31,37)(43,44)(57,59)(62,63), (3,22,19,40,43,42)(4,27,52)(5,12,34,37,36,10)(11,14,13,33,31,35)(15,39)(16,38)(17,44,41,18,21,20)(23,46)(24,45)(25,60,50)(26,28,53)(29,62,54)(30,63,55)(47,61,51)(48,59,58)(49,57,56), (3,21,20,17,43,42)(4,28,52)(5,12,34,31,35,11)(10,14,13,33,37,36)(15,38)(16,39)(18,22,19,40,44,41)(23,45)(24,46)(25,61,50)(26,27,53)(29,62,55)(30,63,54)(47,60,51)(48,59,56)(49,57,58), (3,24,43,42,17,46,21,20)(4,29,59,50)(5,12,11,15,31,35,34,38)(10,16,37,36,33,39,14,13)(18,23,44,41,40,45,22,19)(25,48,62,52)(26,30,57,51)(27,54)(28,55)(47,49,63,53)(56,61)(58,60), (3,23,43,42,17,45,21,20)(4,30,57,50)(5,12,11,16,31,35,34,39)(10,15,37,36,33,38,14,13)(18,24,44,41,40,46,22,19)(25,49,63,52)(26,29,59,51)(27,55)(28,54)(47,48,62,53)(56,60)(58,61), (4,26)(5,14)(12,13)(19,20)(21,22)(25,47)(27,28)(31,37)(35,36)(41,42)(43,44)(54,55)(56,58)(57,59)(60,61)(62,63), (5,14)(15,16)(21,22)(23,24)(27,28)(29,30)(31,37)(38,39)(43,44)(45,46)(48,49)(50,51)(52,53)(54,55)(56,58)(60,61), (4,28)(5,12)(13,14)(15,38)(16,39)(19,22)(20,21)(23,45)(24,46)(25,61)(26,27)(29,48)(30,49)(31,35)(36,37)(41,44)(42,43)(47,60)(50,52)(51,53)(54,63)(55,62)(56,59)(57,58), (4,27,26,28)(5,12,14,13)(15,39)(16,38)(19,21,20,22)(23,46)(24,45)(25,60,47,61)(29,49)(30,48)(31,35,37,36)(41,43,42,44)(50,53)(51,52)(54,63,55,62)(56,59,58,57), (4,30,28)(5,12,39)(13,38,14)(15,37,36)(16,31,35)(19,46,22)(20,45,21)(23,43,42)(24,44,41)(25,49,61)(26,29,27)(47,48,60)(50,58,57)(51,56,59)(52,54,63)(53,55,62), (4,29,28)(5,12,38)(13,39,14)(15,31,35)(16,37,36)(19,45,22)(20,46,21)(23,44,41)(24,43,42)(25,48,61)(26,30,27)(47,49,60)(50,56,59)(51,58,57)(52,55,62)(53,54,63), (5,14)(15,16)(21,22)(23,24)(27,28)(29,30)(31,37)(38,39)(43,44)(45,46)(48,49)(50,51)(52,53)(54,55)(56,58)(60,61), (5,15,31,38)(14,16,37,39)(21,24,43,46)(22,23,44,45)(27,49,60,30)(28,48,61,29)(50,56,52,55)(51,58,53,54), (5,16,31,39)(14,15,37,38)(21,23,43,45)(22,24,44,46)(27,48,60,29)(28,49,61,30)(50,58,52,54)(51,56,53,55));