p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.411C23, C4.1172+ 1+4, C4⋊C4.134D4, Q8⋊D4⋊13C2, C4⋊2Q16⋊29C2, D4⋊D4.4C2, C8.12D4⋊6C2, C8.2D4⋊15C2, C4⋊C8.67C22, C22⋊C4.26D4, C2.31(Q8○D8), D4.7D4⋊31C2, D4.D4⋊12C2, D4.2D4⋊28C2, C4⋊C4.164C23, (C4×C8).114C22, (C2×C8).164C23, (C2×C4).423C24, Q8.D4⋊28C2, C22⋊Q16⋊24C2, C4.SD16⋊17C2, (C2×D8).26C22, C23.295(C2×D4), C4⋊Q8.122C22, C8⋊C4.24C22, C2.48(D4○SD16), (C2×D4).172C23, (C4×D4).110C22, C4⋊D4.46C22, C22⋊C8.58C22, (C2×Q8).160C23, (C4×Q8).107C22, (C2×Q16).28C22, C22⋊Q8.46C22, D4⋊C4.47C22, (C22×C4).311C23, (C2×SD16).39C22, C4.4D4.42C22, C22.683(C22×D4), C42.28C22⋊6C2, C42.7C22⋊15C2, C22.36C24⋊8C2, Q8⋊C4.103C22, (C22×Q8).327C22, C42⋊C2.162C22, C23.38C23⋊17C2, C2.94(C22.29C24), (C2×C4).552(C2×D4), (C2×C4○D4).182C22, SmallGroup(128,1957)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.411C23
G = < a,b,c,d,e | a4=b4=c2=d2=1, e2=b2, ab=ba, cac=dad=a-1b2, eae-1=ab2, cbc=dbd=b-1, be=eb, dcd=bc, ece-1=a2b2c, de=ed >
Subgroups: 388 in 188 conjugacy classes, 84 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4×C8, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42⋊2C2, C4⋊Q8, C2×D8, C2×SD16, C2×Q16, C22×Q8, C2×C4○D4, C42.7C22, Q8⋊D4, D4⋊D4, C22⋊Q16, D4.7D4, D4.D4, C4⋊2Q16, D4.2D4, Q8.D4, C4.SD16, C42.28C22, C8.12D4, C8.2D4, C23.38C23, C22.36C24, C42.411C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, C22.29C24, D4○SD16, Q8○D8, C42.411C23
Character table of C42.411C23
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 8A | 8B | 8C | 8D | 8E | 8F | |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 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 | 0 | -2 | -2 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | -2 | -2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | 2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 4 | -4 | 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 | 0 | -4 | 4 | 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 | -2√2 | 0 | 2√2 | 0 | 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 | 2√2 | 0 | -2√2 | 0 | 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 | 0 | -2√-2 | 0 | 2√-2 | 0 | 0 | complex lifted from D4○SD16 |
| ρ26 | 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 | complex lifted from D4○SD16 |
►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 29 26 50)(2 30 27 51)(3 31 28 52)(4 32 25 49)(5 60 61 53)(6 57 62 54)(7 58 63 55)(8 59 64 56)(9 35 13 40)(10 36 14 37)(11 33 15 38)(12 34 16 39)(17 43 24 48)(18 44 21 45)(19 41 22 46)(20 42 23 47)
(1 58)(2 54)(3 60)(4 56)(5 31)(6 51)(7 29)(8 49)(9 42)(10 46)(11 44)(12 48)(13 47)(14 41)(15 45)(16 43)(17 39)(18 33)(19 37)(20 35)(21 38)(22 36)(23 40)(24 34)(25 59)(26 55)(27 57)(28 53)(30 62)(32 64)(50 63)(52 61)
(2 25)(4 27)(5 60)(6 56)(7 58)(8 54)(10 16)(12 14)(17 46)(18 44)(19 48)(20 42)(21 45)(22 43)(23 47)(24 41)(29 50)(30 32)(31 52)(33 38)(34 36)(35 40)(37 39)(49 51)(53 61)(55 63)(57 64)(59 62)
(1 9 26 13)(2 14 27 10)(3 11 28 15)(4 16 25 12)(5 23 61 20)(6 17 62 24)(7 21 63 18)(8 19 64 22)(29 35 50 40)(30 37 51 36)(31 33 52 38)(32 39 49 34)(41 56 46 59)(42 60 47 53)(43 54 48 57)(44 58 45 55)
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), (1,29,26,50)(2,30,27,51)(3,31,28,52)(4,32,25,49)(5,60,61,53)(6,57,62,54)(7,58,63,55)(8,59,64,56)(9,35,13,40)(10,36,14,37)(11,33,15,38)(12,34,16,39)(17,43,24,48)(18,44,21,45)(19,41,22,46)(20,42,23,47), (1,58)(2,54)(3,60)(4,56)(5,31)(6,51)(7,29)(8,49)(9,42)(10,46)(11,44)(12,48)(13,47)(14,41)(15,45)(16,43)(17,39)(18,33)(19,37)(20,35)(21,38)(22,36)(23,40)(24,34)(25,59)(26,55)(27,57)(28,53)(30,62)(32,64)(50,63)(52,61), (2,25)(4,27)(5,60)(6,56)(7,58)(8,54)(10,16)(12,14)(17,46)(18,44)(19,48)(20,42)(21,45)(22,43)(23,47)(24,41)(29,50)(30,32)(31,52)(33,38)(34,36)(35,40)(37,39)(49,51)(53,61)(55,63)(57,64)(59,62), (1,9,26,13)(2,14,27,10)(3,11,28,15)(4,16,25,12)(5,23,61,20)(6,17,62,24)(7,21,63,18)(8,19,64,22)(29,35,50,40)(30,37,51,36)(31,33,52,38)(32,39,49,34)(41,56,46,59)(42,60,47,53)(43,54,48,57)(44,58,45,55)>;
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), (1,29,26,50)(2,30,27,51)(3,31,28,52)(4,32,25,49)(5,60,61,53)(6,57,62,54)(7,58,63,55)(8,59,64,56)(9,35,13,40)(10,36,14,37)(11,33,15,38)(12,34,16,39)(17,43,24,48)(18,44,21,45)(19,41,22,46)(20,42,23,47), (1,58)(2,54)(3,60)(4,56)(5,31)(6,51)(7,29)(8,49)(9,42)(10,46)(11,44)(12,48)(13,47)(14,41)(15,45)(16,43)(17,39)(18,33)(19,37)(20,35)(21,38)(22,36)(23,40)(24,34)(25,59)(26,55)(27,57)(28,53)(30,62)(32,64)(50,63)(52,61), (2,25)(4,27)(5,60)(6,56)(7,58)(8,54)(10,16)(12,14)(17,46)(18,44)(19,48)(20,42)(21,45)(22,43)(23,47)(24,41)(29,50)(30,32)(31,52)(33,38)(34,36)(35,40)(37,39)(49,51)(53,61)(55,63)(57,64)(59,62), (1,9,26,13)(2,14,27,10)(3,11,28,15)(4,16,25,12)(5,23,61,20)(6,17,62,24)(7,21,63,18)(8,19,64,22)(29,35,50,40)(30,37,51,36)(31,33,52,38)(32,39,49,34)(41,56,46,59)(42,60,47,53)(43,54,48,57)(44,58,45,55) );
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)], [(1,29,26,50),(2,30,27,51),(3,31,28,52),(4,32,25,49),(5,60,61,53),(6,57,62,54),(7,58,63,55),(8,59,64,56),(9,35,13,40),(10,36,14,37),(11,33,15,38),(12,34,16,39),(17,43,24,48),(18,44,21,45),(19,41,22,46),(20,42,23,47)], [(1,58),(2,54),(3,60),(4,56),(5,31),(6,51),(7,29),(8,49),(9,42),(10,46),(11,44),(12,48),(13,47),(14,41),(15,45),(16,43),(17,39),(18,33),(19,37),(20,35),(21,38),(22,36),(23,40),(24,34),(25,59),(26,55),(27,57),(28,53),(30,62),(32,64),(50,63),(52,61)], [(2,25),(4,27),(5,60),(6,56),(7,58),(8,54),(10,16),(12,14),(17,46),(18,44),(19,48),(20,42),(21,45),(22,43),(23,47),(24,41),(29,50),(30,32),(31,52),(33,38),(34,36),(35,40),(37,39),(49,51),(53,61),(55,63),(57,64),(59,62)], [(1,9,26,13),(2,14,27,10),(3,11,28,15),(4,16,25,12),(5,23,61,20),(6,17,62,24),(7,21,63,18),(8,19,64,22),(29,35,50,40),(30,37,51,36),(31,33,52,38),(32,39,49,34),(41,56,46,59),(42,60,47,53),(43,54,48,57),(44,58,45,55)]])
Matrix representation of C42.411C23 ►in GL8(𝔽17)
| 0 | 0 | 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 12 |
| 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 | 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 | 0 | 3 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 14 | 0 | 0 | 0 | 0 |
| 3 | 14 | 0 | 0 | 0 | 0 | 0 | 0 |
| 14 | 14 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 14 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 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 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
G:=sub<GL(8,GF(17))| [0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,0,5,0,12,0,0,0,0,0,0,5,0,12,0,0,0,0,12,0,12,0,0,0,0,0,0,12,0,12],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[0,0,3,14,0,0,0,0,0,0,14,14,0,0,0,0,3,14,0,0,0,0,0,0,14,14,0,0,0,0,0,0,0,0,0,0,14,3,0,0,0,0,0,0,3,3,0,0,0,0,0,0,0,0,14,3,0,0,0,0,0,0,3,3],[1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16],[0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;
C42.411C23 in GAP, Magma, Sage, TeX
C_4^2._{411}C_2^3 % in TeX
G:=Group("C4^2.411C2^3"); // GroupNames label
G:=SmallGroup(128,1957);
// by ID
G=gap.SmallGroup(128,1957);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,219,675,1018,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^4=c^2=d^2=1,e^2=b^2,a*b=b*a,c*a*c=d*a*d=a^-1*b^2,e*a*e^-1=a*b^2,c*b*c=d*b*d=b^-1,b*e=e*b,d*c*d=b*c,e*c*e^-1=a^2*b^2*c,d*e=e*d>;
// generators/relations
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