p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.409C23, C4.1152+ 1+4, C4⋊C4.132D4, C4⋊2Q16⋊28C2, C4⋊Q16⋊10C2, C8.2D4⋊14C2, C4⋊C8.66C22, (C4×C8).75C22, C2.30(Q8○D8), C22⋊C4.24D4, C4⋊C4.162C23, (C2×C8).162C23, (C2×C4).421C24, C22⋊Q16⋊23C2, Q8.D4⋊26C2, D4.7D4.3C2, C23.293(C2×D4), C4⋊Q8.121C22, C8⋊C4.23C22, (C2×D4).170C23, D4⋊C4.2C22, C22⋊C8.56C22, (C2×Q8).158C23, (C2×Q16).73C22, (C4×Q8).106C22, Q8⋊C4.4C22, C22⋊Q8.44C22, (C22×C4).309C23, (C2×SD16).38C22, C4.4D4.41C22, C22.681(C22×D4), C42.C2.25C22, C42.7C22⋊13C2, C42.78C22⋊2C2, C42.30C22⋊4C2, C22.35C24⋊6C2, (C22×Q8).326C22, C42⋊C2.160C22, C2.92(C22.29C24), C23.38C23.15C2, (C2×C4).550(C2×D4), (C2×C4○D4).180C22, SmallGroup(128,1955)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.409C23
G = < a,b,c,d,e | a4=b4=e2=1, c2=d2=b2, ab=ba, cac-1=dad-1=a-1, eae=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece=a2c, de=ed >
Subgroups: 348 in 182 conjugacy classes, 84 normal (32 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4×C8, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C42⋊C2, C4×Q8, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42.C2, C42⋊2C2, C4⋊Q8, C2×SD16, C2×Q16, C22×Q8, C2×C4○D4, C42.7C22, C22⋊Q16, D4.7D4, C4⋊2Q16, Q8.D4, C42.78C22, C42.30C22, C4⋊Q16, C8.2D4, C23.38C23, C22.35C24, C42.409C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, C22.29C24, Q8○D8, C42.409C23
Character table of C42.409C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ9 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ10 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ11 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ12 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ13 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ14 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ15 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ16 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ17 | 2 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | -2 | 0 | -2 | -2 | -2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | -2 | 0 | -2 | -2 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 4 | -4 | 4 | -4 | 0 | 0 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ22 | 4 | -4 | 4 | -4 | 0 | 0 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ23 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | 0 | -2√2 | 0 | 0 | symplectic lifted from Q8○D8, Schur index 2 |
| ρ24 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 0 | 2√2 | 0 | 0 | symplectic lifted from Q8○D8, Schur index 2 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2√2 | 0 | 2√2 | 0 | 0 | 0 | symplectic lifted from Q8○D8, Schur index 2 |
| ρ26 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2√2 | 0 | -2√2 | 0 | 0 | 0 | symplectic lifted from Q8○D8, Schur index 2 |
►On 64 points
Generators in S64
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)
(1 14 51 45)(2 15 52 46)(3 16 49 47)(4 13 50 48)(5 36 56 27)(6 33 53 28)(7 34 54 25)(8 35 55 26)(9 17 21 43)(10 18 22 44)(11 19 23 41)(12 20 24 42)(29 62 37 58)(30 63 38 59)(31 64 39 60)(32 61 40 57)
(1 6 51 53)(2 5 52 56)(3 8 49 55)(4 7 50 54)(9 63 21 59)(10 62 22 58)(11 61 23 57)(12 64 24 60)(13 25 48 34)(14 28 45 33)(15 27 46 36)(16 26 47 35)(17 30 43 38)(18 29 44 37)(19 32 41 40)(20 31 42 39)
(1 24 51 12)(2 23 52 11)(3 22 49 10)(4 21 50 9)(5 40 56 32)(6 39 53 31)(7 38 54 30)(8 37 55 29)(13 17 48 43)(14 20 45 42)(15 19 46 41)(16 18 47 44)(25 59 34 63)(26 58 35 62)(27 57 36 61)(28 60 33 64)
(1 23)(2 12)(3 21)(4 10)(5 62)(6 59)(7 64)(8 57)(9 49)(11 51)(13 18)(14 41)(15 20)(16 43)(17 47)(19 45)(22 50)(24 52)(25 31)(26 40)(27 29)(28 38)(30 33)(32 35)(34 39)(36 37)(42 46)(44 48)(53 63)(54 60)(55 61)(56 58)
G:=sub<Sym(64)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,14,51,45)(2,15,52,46)(3,16,49,47)(4,13,50,48)(5,36,56,27)(6,33,53,28)(7,34,54,25)(8,35,55,26)(9,17,21,43)(10,18,22,44)(11,19,23,41)(12,20,24,42)(29,62,37,58)(30,63,38,59)(31,64,39,60)(32,61,40,57), (1,6,51,53)(2,5,52,56)(3,8,49,55)(4,7,50,54)(9,63,21,59)(10,62,22,58)(11,61,23,57)(12,64,24,60)(13,25,48,34)(14,28,45,33)(15,27,46,36)(16,26,47,35)(17,30,43,38)(18,29,44,37)(19,32,41,40)(20,31,42,39), (1,24,51,12)(2,23,52,11)(3,22,49,10)(4,21,50,9)(5,40,56,32)(6,39,53,31)(7,38,54,30)(8,37,55,29)(13,17,48,43)(14,20,45,42)(15,19,46,41)(16,18,47,44)(25,59,34,63)(26,58,35,62)(27,57,36,61)(28,60,33,64), (1,23)(2,12)(3,21)(4,10)(5,62)(6,59)(7,64)(8,57)(9,49)(11,51)(13,18)(14,41)(15,20)(16,43)(17,47)(19,45)(22,50)(24,52)(25,31)(26,40)(27,29)(28,38)(30,33)(32,35)(34,39)(36,37)(42,46)(44,48)(53,63)(54,60)(55,61)(56,58)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,14,51,45)(2,15,52,46)(3,16,49,47)(4,13,50,48)(5,36,56,27)(6,33,53,28)(7,34,54,25)(8,35,55,26)(9,17,21,43)(10,18,22,44)(11,19,23,41)(12,20,24,42)(29,62,37,58)(30,63,38,59)(31,64,39,60)(32,61,40,57), (1,6,51,53)(2,5,52,56)(3,8,49,55)(4,7,50,54)(9,63,21,59)(10,62,22,58)(11,61,23,57)(12,64,24,60)(13,25,48,34)(14,28,45,33)(15,27,46,36)(16,26,47,35)(17,30,43,38)(18,29,44,37)(19,32,41,40)(20,31,42,39), (1,24,51,12)(2,23,52,11)(3,22,49,10)(4,21,50,9)(5,40,56,32)(6,39,53,31)(7,38,54,30)(8,37,55,29)(13,17,48,43)(14,20,45,42)(15,19,46,41)(16,18,47,44)(25,59,34,63)(26,58,35,62)(27,57,36,61)(28,60,33,64), (1,23)(2,12)(3,21)(4,10)(5,62)(6,59)(7,64)(8,57)(9,49)(11,51)(13,18)(14,41)(15,20)(16,43)(17,47)(19,45)(22,50)(24,52)(25,31)(26,40)(27,29)(28,38)(30,33)(32,35)(34,39)(36,37)(42,46)(44,48)(53,63)(54,60)(55,61)(56,58) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64)], [(1,14,51,45),(2,15,52,46),(3,16,49,47),(4,13,50,48),(5,36,56,27),(6,33,53,28),(7,34,54,25),(8,35,55,26),(9,17,21,43),(10,18,22,44),(11,19,23,41),(12,20,24,42),(29,62,37,58),(30,63,38,59),(31,64,39,60),(32,61,40,57)], [(1,6,51,53),(2,5,52,56),(3,8,49,55),(4,7,50,54),(9,63,21,59),(10,62,22,58),(11,61,23,57),(12,64,24,60),(13,25,48,34),(14,28,45,33),(15,27,46,36),(16,26,47,35),(17,30,43,38),(18,29,44,37),(19,32,41,40),(20,31,42,39)], [(1,24,51,12),(2,23,52,11),(3,22,49,10),(4,21,50,9),(5,40,56,32),(6,39,53,31),(7,38,54,30),(8,37,55,29),(13,17,48,43),(14,20,45,42),(15,19,46,41),(16,18,47,44),(25,59,34,63),(26,58,35,62),(27,57,36,61),(28,60,33,64)], [(1,23),(2,12),(3,21),(4,10),(5,62),(6,59),(7,64),(8,57),(9,49),(11,51),(13,18),(14,41),(15,20),(16,43),(17,47),(19,45),(22,50),(24,52),(25,31),(26,40),(27,29),(28,38),(30,33),(32,35),(34,39),(36,37),(42,46),(44,48),(53,63),(54,60),(55,61),(56,58)]])
Matrix representation of C42.409C23 ►in GL8(𝔽17)
| 12 | 10 | 12 | 0 | 0 | 0 | 0 | 0 |
| 12 | 5 | 0 | 12 | 0 | 0 | 0 | 0 |
| 5 | 0 | 5 | 7 | 0 | 0 | 0 | 0 |
| 0 | 5 | 5 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 7 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 10 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 10 |
| 1 | 15 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 13 | 13 | 4 | 0 | 0 | 0 | 0 |
| 15 | 0 | 15 | 4 | 0 | 0 | 0 | 0 |
| 13 | 4 | 0 | 13 | 0 | 0 | 0 | 0 |
| 15 | 4 | 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 14 |
| 0 | 0 | 0 | 0 | 14 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 3 | 0 | 0 |
| 12 | 0 | 12 | 10 | 0 | 0 | 0 | 0 |
| 12 | 5 | 0 | 5 | 0 | 0 | 0 | 0 |
| 12 | 10 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 12 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 10 | 0 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 7 |
| 0 | 0 | 0 | 0 | 10 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 0 | 7 | 16 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(8,GF(17))| [12,12,5,0,0,0,0,0,10,5,0,5,0,0,0,0,12,0,5,5,0,0,0,0,0,12,7,12,0,0,0,0,0,0,0,0,7,0,0,1,0,0,0,0,0,7,16,0,0,0,0,0,0,16,10,0,0,0,0,0,1,0,0,10],[1,1,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[0,15,13,15,0,0,0,0,13,0,4,4,0,0,0,0,13,15,0,15,0,0,0,0,4,4,13,0,0,0,0,0,0,0,0,0,0,0,14,14,0,0,0,0,0,0,14,3,0,0,0,0,3,3,0,0,0,0,0,0,3,14,0,0],[12,12,12,0,0,0,0,0,0,5,10,5,0,0,0,0,12,0,12,12,0,0,0,0,10,5,0,5,0,0,0,0,0,0,0,0,0,16,10,0,0,0,0,0,16,0,0,7,0,0,0,0,10,0,0,16,0,0,0,0,0,7,16,0],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;
C42.409C23 in GAP, Magma, Sage, TeX
C_4^2._{409}C_2^3 % in TeX
G:=Group("C4^2.409C2^3"); // GroupNames label
G:=SmallGroup(128,1955);
// by ID
G=gap.SmallGroup(128,1955);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,219,352,675,1018,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,e*a*e=a*b^2,c*b*c^-1=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=b*c,e*c*e=a^2*c,d*e=e*d>;
// generators/relations
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