metabelian, supersoluble, monomial, A-group
Aliases: C32:4F7, C3:(C3:F7), C21:1(C3xS3), (C3xC21):4C6, C3:D21:3C3, C7:(C3xC3:S3), C7:C3:(C3:S3), (C3xC7:C3):3S3, (C32xC7:C3):2C2, SmallGroup(378,51)
Series: Derived ►Chief ►Lower central ►Upper central
| C3xC21 — C32:4F7 |
Generators and relations for C32:4F7
G = < a,b,c,d | a3=b3=c7=d6=1, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c5 >
Subgroups: 540 in 64 conjugacy classes, 20 normal (8 characteristic)
C1, C2, C3, C3, S3, C6, C7, C32, C32, D7, C3xS3, C3:S3, C7:C3, C7:C3, C21, C33, F7, D21, C3xC3:S3, C3xC7:C3, C3xC7:C3, C3xC21, C3:F7, C3:D21, C32xC7:C3, C32:4F7
Quotients: C1, C2, C3, S3, C6, C3xS3, C3:S3, F7, C3xC3:S3, C3:F7, C32:4F7
Character table of C32:4F7
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 3I | 3J | 3K | 3L | 3M | 3N | 6A | 6B | 7 | 21A | 21B | 21C | 21D | 21E | 21F | 21G | 21H | |
| size | 1 | 63 | 2 | 2 | 2 | 2 | 7 | 7 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 14 | 63 | 63 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | |
| 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 | 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 | 1 | linear of order 2 |
| ρ3 | 1 | -1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ4 | 1 | -1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ7 | 2 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | 0 | 0 | 2 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | orthogonal lifted from S3 |
| ρ8 | 2 | 0 | -1 | -1 | 2 | -1 | 2 | 2 | -1 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | 0 | 0 | 2 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ9 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | 0 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 0 | -1 | 2 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | 2 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 0 | -1 | 2 | -1 | -1 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | ζ6 | ζ6 | ζ65 | -1+√-3 | ζ65 | -1-√-3 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | complex lifted from C3xS3 |
| ρ12 | 2 | 0 | -1 | -1 | -1 | 2 | -1+√-3 | -1-√-3 | -1-√-3 | ζ65 | ζ65 | -1+√-3 | ζ6 | ζ6 | ζ6 | ζ65 | 0 | 0 | 2 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | complex lifted from C3xS3 |
| ρ13 | 2 | 0 | 2 | -1 | -1 | -1 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | -1+√-3 | ζ65 | -1-√-3 | ζ6 | ζ6 | ζ65 | 0 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | -1 | complex lifted from C3xS3 |
| ρ14 | 2 | 0 | -1 | 2 | -1 | -1 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | ζ65 | ζ65 | ζ6 | -1-√-3 | ζ6 | -1+√-3 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | complex lifted from C3xS3 |
| ρ15 | 2 | 0 | -1 | -1 | 2 | -1 | -1-√-3 | -1+√-3 | ζ65 | -1-√-3 | ζ6 | ζ6 | ζ65 | ζ65 | -1+√-3 | ζ6 | 0 | 0 | 2 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | complex lifted from C3xS3 |
| ρ16 | 2 | 0 | -1 | -1 | 2 | -1 | -1+√-3 | -1-√-3 | ζ6 | -1+√-3 | ζ65 | ζ65 | ζ6 | ζ6 | -1-√-3 | ζ65 | 0 | 0 | 2 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | complex lifted from C3xS3 |
| ρ17 | 2 | 0 | 2 | -1 | -1 | -1 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | -1-√-3 | ζ6 | -1+√-3 | ζ65 | ζ65 | ζ6 | 0 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | -1 | complex lifted from C3xS3 |
| ρ18 | 2 | 0 | -1 | -1 | -1 | 2 | -1-√-3 | -1+√-3 | -1+√-3 | ζ6 | ζ6 | -1-√-3 | ζ65 | ζ65 | ζ65 | ζ6 | 0 | 0 | 2 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | complex lifted from C3xS3 |
| ρ19 | 6 | 0 | 6 | 6 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F7 |
| ρ20 | 6 | 0 | -3 | -3 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1+√21/2 | 1+√21/2 | -1 | -1 | 1-√21/2 | 1-√21/2 | 1+√21/2 | 1-√21/2 | orthogonal lifted from C3:F7 |
| ρ21 | 6 | 0 | -3 | -3 | -3 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1+√21/2 | -1 | 1+√21/2 | 1-√21/2 | 1-√21/2 | 1+√21/2 | 1-√21/2 | -1 | orthogonal lifted from C3:F7 |
| ρ22 | 6 | 0 | 6 | -3 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1-√21/2 | 1+√21/2 | 1+√21/2 | 1-√21/2 | 1+√21/2 | -1 | -1 | 1-√21/2 | orthogonal lifted from C3:F7 |
| ρ23 | 6 | 0 | -3 | -3 | -3 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1-√21/2 | -1 | 1-√21/2 | 1+√21/2 | 1+√21/2 | 1-√21/2 | 1+√21/2 | -1 | orthogonal lifted from C3:F7 |
| ρ24 | 6 | 0 | 6 | -3 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1+√21/2 | 1-√21/2 | 1-√21/2 | 1+√21/2 | 1-√21/2 | -1 | -1 | 1+√21/2 | orthogonal lifted from C3:F7 |
| ρ25 | 6 | 0 | -3 | -3 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1-√21/2 | 1-√21/2 | -1 | -1 | 1+√21/2 | 1+√21/2 | 1-√21/2 | 1+√21/2 | orthogonal lifted from C3:F7 |
| ρ26 | 6 | 0 | -3 | 6 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1-√21/2 | 1+√21/2 | 1-√21/2 | -1 | 1-√21/2 | 1+√21/2 | 1+√21/2 | orthogonal lifted from C3:F7 |
| ρ27 | 6 | 0 | -3 | 6 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1+√21/2 | 1-√21/2 | 1+√21/2 | -1 | 1+√21/2 | 1-√21/2 | 1-√21/2 | orthogonal lifted from C3:F7 |
►On 63 points
Generators in S63
(1 43 22)(2 44 23)(3 45 24)(4 46 25)(5 47 26)(6 48 27)(7 49 28)(8 50 29)(9 51 30)(10 52 31)(11 53 32)(12 54 33)(13 55 34)(14 56 35)(15 57 36)(16 58 37)(17 59 38)(18 60 39)(19 61 40)(20 62 41)(21 63 42)
(1 15 8)(2 16 9)(3 17 10)(4 18 11)(5 19 12)(6 20 13)(7 21 14)(22 36 29)(23 37 30)(24 38 31)(25 39 32)(26 40 33)(27 41 34)(28 42 35)(43 57 50)(44 58 51)(45 59 52)(46 60 53)(47 61 54)(48 62 55)(49 63 56)
(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)
(2 4 3 7 5 6)(8 15)(9 18 10 21 12 20)(11 17 14 19 13 16)(22 43)(23 46 24 49 26 48)(25 45 28 47 27 44)(29 57)(30 60 31 63 33 62)(32 59 35 61 34 58)(36 50)(37 53 38 56 40 55)(39 52 42 54 41 51)
G:=sub<Sym(63)| (1,43,22)(2,44,23)(3,45,24)(4,46,25)(5,47,26)(6,48,27)(7,49,28)(8,50,29)(9,51,30)(10,52,31)(11,53,32)(12,54,33)(13,55,34)(14,56,35)(15,57,36)(16,58,37)(17,59,38)(18,60,39)(19,61,40)(20,62,41)(21,63,42), (1,15,8)(2,16,9)(3,17,10)(4,18,11)(5,19,12)(6,20,13)(7,21,14)(22,36,29)(23,37,30)(24,38,31)(25,39,32)(26,40,33)(27,41,34)(28,42,35)(43,57,50)(44,58,51)(45,59,52)(46,60,53)(47,61,54)(48,62,55)(49,63,56), (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), (2,4,3,7,5,6)(8,15)(9,18,10,21,12,20)(11,17,14,19,13,16)(22,43)(23,46,24,49,26,48)(25,45,28,47,27,44)(29,57)(30,60,31,63,33,62)(32,59,35,61,34,58)(36,50)(37,53,38,56,40,55)(39,52,42,54,41,51)>;
G:=Group( (1,43,22)(2,44,23)(3,45,24)(4,46,25)(5,47,26)(6,48,27)(7,49,28)(8,50,29)(9,51,30)(10,52,31)(11,53,32)(12,54,33)(13,55,34)(14,56,35)(15,57,36)(16,58,37)(17,59,38)(18,60,39)(19,61,40)(20,62,41)(21,63,42), (1,15,8)(2,16,9)(3,17,10)(4,18,11)(5,19,12)(6,20,13)(7,21,14)(22,36,29)(23,37,30)(24,38,31)(25,39,32)(26,40,33)(27,41,34)(28,42,35)(43,57,50)(44,58,51)(45,59,52)(46,60,53)(47,61,54)(48,62,55)(49,63,56), (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), (2,4,3,7,5,6)(8,15)(9,18,10,21,12,20)(11,17,14,19,13,16)(22,43)(23,46,24,49,26,48)(25,45,28,47,27,44)(29,57)(30,60,31,63,33,62)(32,59,35,61,34,58)(36,50)(37,53,38,56,40,55)(39,52,42,54,41,51) );
G=PermutationGroup([[(1,43,22),(2,44,23),(3,45,24),(4,46,25),(5,47,26),(6,48,27),(7,49,28),(8,50,29),(9,51,30),(10,52,31),(11,53,32),(12,54,33),(13,55,34),(14,56,35),(15,57,36),(16,58,37),(17,59,38),(18,60,39),(19,61,40),(20,62,41),(21,63,42)], [(1,15,8),(2,16,9),(3,17,10),(4,18,11),(5,19,12),(6,20,13),(7,21,14),(22,36,29),(23,37,30),(24,38,31),(25,39,32),(26,40,33),(27,41,34),(28,42,35),(43,57,50),(44,58,51),(45,59,52),(46,60,53),(47,61,54),(48,62,55),(49,63,56)], [(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)], [(2,4,3,7,5,6),(8,15),(9,18,10,21,12,20),(11,17,14,19,13,16),(22,43),(23,46,24,49,26,48),(25,45,28,47,27,44),(29,57),(30,60,31,63,33,62),(32,59,35,61,34,58),(36,50),(37,53,38,56,40,55),(39,52,42,54,41,51)]])
Matrix representation of C32:4F7 ►in GL8(F43)
| 1 | 36 | 0 | 0 | 0 | 0 | 0 | 0 |
| 25 | 41 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 5 | 5 | 0 | 5 | 0 |
| 0 | 0 | 0 | 2 | 5 | 5 | 0 | 5 |
| 0 | 0 | 38 | 38 | 40 | 0 | 0 | 38 |
| 0 | 0 | 5 | 0 | 0 | 2 | 5 | 5 |
| 0 | 0 | 38 | 0 | 38 | 38 | 40 | 0 |
| 0 | 0 | 0 | 38 | 0 | 38 | 38 | 40 |
| 41 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 18 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 42 | 42 | 42 | 42 | 42 | 42 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 37 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 22 | 6 | 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 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 42 | 42 | 42 | 42 | 42 | 42 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(8,GF(43))| [1,25,0,0,0,0,0,0,36,41,0,0,0,0,0,0,0,0,2,0,38,5,38,0,0,0,5,2,38,0,0,38,0,0,5,5,40,0,38,0,0,0,0,5,0,2,38,38,0,0,5,0,0,5,40,38,0,0,0,5,38,5,0,40],[41,18,0,0,0,0,0,0,7,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,42,1,0,0,0,0,0,0,42,0,1,0,0,0,0,0,42,0,0,1,0,0,0,0,42,0,0,0,1,0,0,0,42,0,0,0,0,1,0,0,42,0,0,0,0,0],[37,22,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,1,0,0,0,42,0,0,0,0,0,0,1,42,0,0,0,0,0,0,0,42,0,0,0,0,0,1,0,42,0,0,0,0,0,0,0,42,1,0,0,0,1,0,0,42,0] >;
C32:4F7 in GAP, Magma, Sage, TeX
C_3^2\rtimes_4F_7
% in TeX
G:=Group("C3^2:4F7"); // GroupNames label
G:=SmallGroup(378,51);
// by ID
G=gap.SmallGroup(378,51);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-7,182,723,8104,1359]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^7=d^6=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations
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