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## G = D9×C7⋊C3order 378 = 2·33·7

### Direct product of D9 and C7⋊C3

Aliases: D9×C7⋊C3, C638C6, C72(C3×D9), (C7×D9)⋊4C3, C21.8(C3×S3), (C9×C7⋊C3)⋊2C2, C95(C2×C7⋊C3), C3.3(S3×C7⋊C3), (C3×C7⋊C3).6S3, SmallGroup(378,15)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C63 — D9×C7⋊C3
 Chief series C1 — C3 — C21 — C63 — C9×C7⋊C3 — D9×C7⋊C3
 Lower central C63 — D9×C7⋊C3
 Upper central C1

Generators and relations for D9×C7⋊C3
G = < a,b,c,d | a9=b2=c7=d3=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

Character table of D9×C7⋊C3

 class 1 2 3A 3B 3C 3D 3E 6A 6B 7A 7B 9A 9B 9C 9D 9E 9F 9G 9H 9I 14A 14B 21A 21B 63A 63B 63C 63D 63E 63F size 1 9 2 7 7 14 14 63 63 3 3 2 2 2 14 14 14 14 14 14 27 27 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 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 1 1 1 linear of order 2 ρ3 1 -1 1 ζ32 ζ3 ζ3 ζ32 ζ65 ζ6 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 -1 -1 1 1 1 1 1 1 1 1 linear of order 6 ρ4 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ5 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 1 1 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 1 1 1 1 1 1 1 1 1 1 linear of order 3 ρ6 1 -1 1 ζ3 ζ32 ζ32 ζ3 ζ6 ζ65 1 1 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 -1 -1 1 1 1 1 1 1 1 1 linear of order 6 ρ7 2 0 2 2 2 2 2 0 0 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 0 0 2 2 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ8 2 0 -1 2 2 -1 -1 0 0 2 2 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 0 0 -1 -1 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D9 ρ9 2 0 -1 2 2 -1 -1 0 0 2 2 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 0 0 -1 -1 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D9 ρ10 2 0 -1 2 2 -1 -1 0 0 2 2 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 0 0 -1 -1 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D9 ρ11 2 0 2 -1+√-3 -1-√-3 -1-√-3 -1+√-3 0 0 2 2 -1 -1 -1 ζ65 ζ6 ζ6 ζ65 ζ65 ζ6 0 0 2 2 -1 -1 -1 -1 -1 -1 complex lifted from C3×S3 ρ12 2 0 -1 -1+√-3 -1-√-3 ζ6 ζ65 0 0 2 2 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ94+ζ92 ζ98+ζ94 ζ92+ζ9 ζ98+ζ97 ζ95+ζ9 ζ97+ζ95 0 0 -1 -1 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 complex lifted from C3×D9 ρ13 2 0 2 -1-√-3 -1+√-3 -1+√-3 -1-√-3 0 0 2 2 -1 -1 -1 ζ6 ζ65 ζ65 ζ6 ζ6 ζ65 0 0 2 2 -1 -1 -1 -1 -1 -1 complex lifted from C3×S3 ρ14 2 0 -1 -1-√-3 -1+√-3 ζ65 ζ6 0 0 2 2 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ98+ζ94 ζ98+ζ97 ζ94+ζ92 ζ97+ζ95 ζ92+ζ9 ζ95+ζ9 0 0 -1 -1 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 complex lifted from C3×D9 ρ15 2 0 -1 -1-√-3 -1+√-3 ζ65 ζ6 0 0 2 2 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ92+ζ9 ζ94+ζ92 ζ95+ζ9 ζ98+ζ94 ζ97+ζ95 ζ98+ζ97 0 0 -1 -1 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 complex lifted from C3×D9 ρ16 2 0 -1 -1+√-3 -1-√-3 ζ6 ζ65 0 0 2 2 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ95+ζ9 ζ92+ζ9 ζ97+ζ95 ζ94+ζ92 ζ98+ζ97 ζ98+ζ94 0 0 -1 -1 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 complex lifted from C3×D9 ρ17 2 0 -1 -1-√-3 -1+√-3 ζ65 ζ6 0 0 2 2 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ97+ζ95 ζ95+ζ9 ζ98+ζ97 ζ92+ζ9 ζ98+ζ94 ζ94+ζ92 0 0 -1 -1 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 complex lifted from C3×D9 ρ18 2 0 -1 -1+√-3 -1-√-3 ζ6 ζ65 0 0 2 2 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ98+ζ97 ζ97+ζ95 ζ98+ζ94 ζ95+ζ9 ζ94+ζ92 ζ92+ζ9 0 0 -1 -1 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 complex lifted from C3×D9 ρ19 3 -3 3 0 0 0 0 0 0 -1+√-7/2 -1-√-7/2 3 3 3 0 0 0 0 0 0 1+√-7/2 1-√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 complex lifted from C2×C7⋊C3 ρ20 3 3 3 0 0 0 0 0 0 -1+√-7/2 -1-√-7/2 3 3 3 0 0 0 0 0 0 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 complex lifted from C7⋊C3 ρ21 3 3 3 0 0 0 0 0 0 -1-√-7/2 -1+√-7/2 3 3 3 0 0 0 0 0 0 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 complex lifted from C7⋊C3 ρ22 3 -3 3 0 0 0 0 0 0 -1-√-7/2 -1+√-7/2 3 3 3 0 0 0 0 0 0 1-√-7/2 1+√-7/2 -1-√-7/2 -1+√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 complex lifted from C2×C7⋊C3 ρ23 6 0 6 0 0 0 0 0 0 -1+√-7 -1-√-7 -3 -3 -3 0 0 0 0 0 0 0 0 -1+√-7 -1-√-7 1-√-7/2 1+√-7/2 1+√-7/2 1-√-7/2 1-√-7/2 1+√-7/2 complex lifted from S3×C7⋊C3 ρ24 6 0 6 0 0 0 0 0 0 -1-√-7 -1+√-7 -3 -3 -3 0 0 0 0 0 0 0 0 -1-√-7 -1+√-7 1+√-7/2 1-√-7/2 1-√-7/2 1+√-7/2 1+√-7/2 1-√-7/2 complex lifted from S3×C7⋊C3 ρ25 6 0 -3 0 0 0 0 0 0 -1+√-7 -1-√-7 3ζ97+3ζ92 3ζ98+3ζ9 3ζ95+3ζ94 0 0 0 0 0 0 0 0 1-√-7/2 1+√-7/2 ζ97ζ74+ζ97ζ72+ζ97ζ7+ζ92ζ74+ζ92ζ72+ζ92ζ7 ζ97ζ76+ζ97ζ75+ζ97ζ73+ζ92ζ76+ζ92ζ75+ζ92ζ73 ζ95ζ76+ζ95ζ75+ζ95ζ73+ζ94ζ76+ζ94ζ75+ζ94ζ73 ζ98ζ74+ζ98ζ72+ζ98ζ7+ζ9ζ74+ζ9ζ72+ζ9ζ7 ζ95ζ74+ζ95ζ72+ζ95ζ7+ζ94ζ74+ζ94ζ72+ζ94ζ7 ζ98ζ76+ζ98ζ75+ζ98ζ73+ζ9ζ76+ζ9ζ75+ζ9ζ73 complex faithful ρ26 6 0 -3 0 0 0 0 0 0 -1+√-7 -1-√-7 3ζ95+3ζ94 3ζ97+3ζ92 3ζ98+3ζ9 0 0 0 0 0 0 0 0 1-√-7/2 1+√-7/2 ζ95ζ74+ζ95ζ72+ζ95ζ7+ζ94ζ74+ζ94ζ72+ζ94ζ7 ζ95ζ76+ζ95ζ75+ζ95ζ73+ζ94ζ76+ζ94ζ75+ζ94ζ73 ζ98ζ76+ζ98ζ75+ζ98ζ73+ζ9ζ76+ζ9ζ75+ζ9ζ73 ζ97ζ74+ζ97ζ72+ζ97ζ7+ζ92ζ74+ζ92ζ72+ζ92ζ7 ζ98ζ74+ζ98ζ72+ζ98ζ7+ζ9ζ74+ζ9ζ72+ζ9ζ7 ζ97ζ76+ζ97ζ75+ζ97ζ73+ζ92ζ76+ζ92ζ75+ζ92ζ73 complex faithful ρ27 6 0 -3 0 0 0 0 0 0 -1+√-7 -1-√-7 3ζ98+3ζ9 3ζ95+3ζ94 3ζ97+3ζ92 0 0 0 0 0 0 0 0 1-√-7/2 1+√-7/2 ζ98ζ74+ζ98ζ72+ζ98ζ7+ζ9ζ74+ζ9ζ72+ζ9ζ7 ζ98ζ76+ζ98ζ75+ζ98ζ73+ζ9ζ76+ζ9ζ75+ζ9ζ73 ζ97ζ76+ζ97ζ75+ζ97ζ73+ζ92ζ76+ζ92ζ75+ζ92ζ73 ζ95ζ74+ζ95ζ72+ζ95ζ7+ζ94ζ74+ζ94ζ72+ζ94ζ7 ζ97ζ74+ζ97ζ72+ζ97ζ7+ζ92ζ74+ζ92ζ72+ζ92ζ7 ζ95ζ76+ζ95ζ75+ζ95ζ73+ζ94ζ76+ζ94ζ75+ζ94ζ73 complex faithful ρ28 6 0 -3 0 0 0 0 0 0 -1-√-7 -1+√-7 3ζ95+3ζ94 3ζ97+3ζ92 3ζ98+3ζ9 0 0 0 0 0 0 0 0 1+√-7/2 1-√-7/2 ζ95ζ76+ζ95ζ75+ζ95ζ73+ζ94ζ76+ζ94ζ75+ζ94ζ73 ζ95ζ74+ζ95ζ72+ζ95ζ7+ζ94ζ74+ζ94ζ72+ζ94ζ7 ζ98ζ74+ζ98ζ72+ζ98ζ7+ζ9ζ74+ζ9ζ72+ζ9ζ7 ζ97ζ76+ζ97ζ75+ζ97ζ73+ζ92ζ76+ζ92ζ75+ζ92ζ73 ζ98ζ76+ζ98ζ75+ζ98ζ73+ζ9ζ76+ζ9ζ75+ζ9ζ73 ζ97ζ74+ζ97ζ72+ζ97ζ7+ζ92ζ74+ζ92ζ72+ζ92ζ7 complex faithful ρ29 6 0 -3 0 0 0 0 0 0 -1-√-7 -1+√-7 3ζ98+3ζ9 3ζ95+3ζ94 3ζ97+3ζ92 0 0 0 0 0 0 0 0 1+√-7/2 1-√-7/2 ζ98ζ76+ζ98ζ75+ζ98ζ73+ζ9ζ76+ζ9ζ75+ζ9ζ73 ζ98ζ74+ζ98ζ72+ζ98ζ7+ζ9ζ74+ζ9ζ72+ζ9ζ7 ζ97ζ74+ζ97ζ72+ζ97ζ7+ζ92ζ74+ζ92ζ72+ζ92ζ7 ζ95ζ76+ζ95ζ75+ζ95ζ73+ζ94ζ76+ζ94ζ75+ζ94ζ73 ζ97ζ76+ζ97ζ75+ζ97ζ73+ζ92ζ76+ζ92ζ75+ζ92ζ73 ζ95ζ74+ζ95ζ72+ζ95ζ7+ζ94ζ74+ζ94ζ72+ζ94ζ7 complex faithful ρ30 6 0 -3 0 0 0 0 0 0 -1-√-7 -1+√-7 3ζ97+3ζ92 3ζ98+3ζ9 3ζ95+3ζ94 0 0 0 0 0 0 0 0 1+√-7/2 1-√-7/2 ζ97ζ76+ζ97ζ75+ζ97ζ73+ζ92ζ76+ζ92ζ75+ζ92ζ73 ζ97ζ74+ζ97ζ72+ζ97ζ7+ζ92ζ74+ζ92ζ72+ζ92ζ7 ζ95ζ74+ζ95ζ72+ζ95ζ7+ζ94ζ74+ζ94ζ72+ζ94ζ7 ζ98ζ76+ζ98ζ75+ζ98ζ73+ζ9ζ76+ζ9ζ75+ζ9ζ73 ζ95ζ76+ζ95ζ75+ζ95ζ73+ζ94ζ76+ζ94ζ75+ζ94ζ73 ζ98ζ74+ζ98ζ72+ζ98ζ7+ζ9ζ74+ζ9ζ72+ζ9ζ7 complex faithful

Smallest permutation representation of D9×C7⋊C3
On 63 points
Generators in S63
(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)
(1 9)(2 8)(3 7)(4 6)(10 17)(11 16)(12 15)(13 14)(19 25)(20 24)(21 23)(26 27)(28 36)(29 35)(30 34)(31 33)(37 42)(38 41)(39 40)(43 45)(46 54)(47 53)(48 52)(49 51)(55 56)(57 63)(58 62)(59 61)
(1 28 46 27 56 14 40)(2 29 47 19 57 15 41)(3 30 48 20 58 16 42)(4 31 49 21 59 17 43)(5 32 50 22 60 18 44)(6 33 51 23 61 10 45)(7 34 52 24 62 11 37)(8 35 53 25 63 12 38)(9 36 54 26 55 13 39)
(10 23 45)(11 24 37)(12 25 38)(13 26 39)(14 27 40)(15 19 41)(16 20 42)(17 21 43)(18 22 44)(28 46 56)(29 47 57)(30 48 58)(31 49 59)(32 50 60)(33 51 61)(34 52 62)(35 53 63)(36 54 55)

G:=sub<Sym(63)| (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), (1,9)(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,25)(20,24)(21,23)(26,27)(28,36)(29,35)(30,34)(31,33)(37,42)(38,41)(39,40)(43,45)(46,54)(47,53)(48,52)(49,51)(55,56)(57,63)(58,62)(59,61), (1,28,46,27,56,14,40)(2,29,47,19,57,15,41)(3,30,48,20,58,16,42)(4,31,49,21,59,17,43)(5,32,50,22,60,18,44)(6,33,51,23,61,10,45)(7,34,52,24,62,11,37)(8,35,53,25,63,12,38)(9,36,54,26,55,13,39), (10,23,45)(11,24,37)(12,25,38)(13,26,39)(14,27,40)(15,19,41)(16,20,42)(17,21,43)(18,22,44)(28,46,56)(29,47,57)(30,48,58)(31,49,59)(32,50,60)(33,51,61)(34,52,62)(35,53,63)(36,54,55)>;

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), (1,9)(2,8)(3,7)(4,6)(10,17)(11,16)(12,15)(13,14)(19,25)(20,24)(21,23)(26,27)(28,36)(29,35)(30,34)(31,33)(37,42)(38,41)(39,40)(43,45)(46,54)(47,53)(48,52)(49,51)(55,56)(57,63)(58,62)(59,61), (1,28,46,27,56,14,40)(2,29,47,19,57,15,41)(3,30,48,20,58,16,42)(4,31,49,21,59,17,43)(5,32,50,22,60,18,44)(6,33,51,23,61,10,45)(7,34,52,24,62,11,37)(8,35,53,25,63,12,38)(9,36,54,26,55,13,39), (10,23,45)(11,24,37)(12,25,38)(13,26,39)(14,27,40)(15,19,41)(16,20,42)(17,21,43)(18,22,44)(28,46,56)(29,47,57)(30,48,58)(31,49,59)(32,50,60)(33,51,61)(34,52,62)(35,53,63)(36,54,55) );

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)], [(1,9),(2,8),(3,7),(4,6),(10,17),(11,16),(12,15),(13,14),(19,25),(20,24),(21,23),(26,27),(28,36),(29,35),(30,34),(31,33),(37,42),(38,41),(39,40),(43,45),(46,54),(47,53),(48,52),(49,51),(55,56),(57,63),(58,62),(59,61)], [(1,28,46,27,56,14,40),(2,29,47,19,57,15,41),(3,30,48,20,58,16,42),(4,31,49,21,59,17,43),(5,32,50,22,60,18,44),(6,33,51,23,61,10,45),(7,34,52,24,62,11,37),(8,35,53,25,63,12,38),(9,36,54,26,55,13,39)], [(10,23,45),(11,24,37),(12,25,38),(13,26,39),(14,27,40),(15,19,41),(16,20,42),(17,21,43),(18,22,44),(28,46,56),(29,47,57),(30,48,58),(31,49,59),(32,50,60),(33,51,61),(34,52,62),(35,53,63),(36,54,55)]])

Matrix representation of D9×C7⋊C3 in GL5(𝔽127)

 96 9 0 0 0 118 105 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 105 31 0 0 0 9 22 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 126 104 1 0 0 0 104 1 0 0 126 105 1
,
 107 0 0 0 0 0 107 0 0 0 0 0 105 1 23 0 0 1 0 0 0 0 1 1 22

G:=sub<GL(5,GF(127))| [96,118,0,0,0,9,105,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[105,9,0,0,0,31,22,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,126,0,126,0,0,104,104,105,0,0,1,1,1],[107,0,0,0,0,0,107,0,0,0,0,0,105,1,1,0,0,1,0,1,0,0,23,0,22] >;

D9×C7⋊C3 in GAP, Magma, Sage, TeX

D_9\times C_7\rtimes C_3
% in TeX

G:=Group("D9xC7:C3");
// GroupNames label

G:=SmallGroup(378,15);
// by ID

G=gap.SmallGroup(378,15);
# by ID

G:=PCGroup([5,-2,-3,-3,-7,-3,2072,642,368,6304]);
// Polycyclic

G:=Group<a,b,c,d|a^9=b^2=c^7=d^3=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations

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