metabelian, supersoluble, monomial, A-group
Aliases: C32⋊4F7, C3⋊(C3⋊F7), C21⋊1(C3×S3), (C3×C21)⋊4C6, C3⋊D21⋊3C3, C7⋊(C3×C3⋊S3), C7⋊C3⋊(C3⋊S3), (C3×C7⋊C3)⋊3S3, (C32×C7⋊C3)⋊2C2, SmallGroup(378,51)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C21 — 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, C3×S3, C3⋊S3, C7⋊C3, C7⋊C3, C21, C33, F7, D21, C3×C3⋊S3, C3×C7⋊C3, C3×C7⋊C3, C3×C21, C3⋊F7, C3⋊D21, C32×C7⋊C3, C32⋊4F7
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, F7, C3×C3⋊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 C3×S3 |
| 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 C3×S3 |
| 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 C3×S3 |
| 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 C3×S3 |
| 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 C3×S3 |
| 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 C3×S3 |
| 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 C3×S3 |
| 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 C3×S3 |
| 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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