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G = D7×D9  order 252 = 22·32·7

Direct product of D7 and D9

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D7×D9, D63⋊C2, C9⋊1D14, C7⋊1D18, C63⋊C22, C21.D6, (C7×D9)⋊C2, (C9×D7)⋊C2, C3.(S3×D7), (C3×D7).S3, SmallGroup(252,8)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C63 — D7×D9
Chief series
C1 — C3 — C21 — C63 — C9×D7 — D7×D9
Lower central
C63 — D7×D9
Upper central
C1

Generators and relations for D7×D9
 G = < a,b,c,d | a7=b2=c9=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

C1
7C2
9C2
63C2
C3
C7
63C22
3S3
7C6
21S3
C9
D7
9C14
9D7
C21
21D6
D9
7C18
7D9
9D14
C3×D7
3S3×C7
3D21
C63
7D18
3S3×D7
D63
C7×D9
C9×D7
D7×D9

Character table of D7×D9

 class 12A2B2C367A7B7C9A9B9C14A14B14C18A18B18C21A21B21C63A63B63C63D63E63F63G63H63I
 size 17963214222222181818141414444444444444
ρ1111111111111111111111111111111    trivial
ρ21-11-11-1111111111-1-1-1111111111111    linear of order 2
ρ311-1-111111111-1-1-1111111111111111    linear of order 2
ρ41-1-111-1111111-1-1-1-1-1-1111111111111    linear of order 2
ρ5220022222-1-1-1000-1-1-1222-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ62-2002-2222-1-1-1000111222-1-1-1-1-1-1-1-1-1    orthogonal lifted from D6
ρ720-2020ζ75+ζ72ζ74+ζ73ζ76+ζ7222-ζ74-ζ73-ζ75-ζ72-ζ76-ζ7000ζ75+ζ72ζ76+ζ7ζ74+ζ73ζ75+ζ72ζ75+ζ72ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ75+ζ72    orthogonal lifted from D14
ρ820-2020ζ74+ζ73ζ76+ζ7ζ75+ζ72222-ζ76-ζ7-ζ74-ζ73-ζ75-ζ72000ζ74+ζ73ζ75+ζ72ζ76+ζ7ζ74+ζ73ζ74+ζ73ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ74+ζ73    orthogonal lifted from D14
ρ920-2020ζ76+ζ7ζ75+ζ72ζ74+ζ73222-ζ75-ζ72-ζ76-ζ7-ζ74-ζ73000ζ76+ζ7ζ74+ζ73ζ75+ζ72ζ76+ζ7ζ76+ζ7ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ76+ζ7    orthogonal lifted from D14
ρ10202020ζ76+ζ7ζ75+ζ72ζ74+ζ73222ζ75+ζ72ζ76+ζ7ζ74+ζ73000ζ76+ζ7ζ74+ζ73ζ75+ζ72ζ76+ζ7ζ76+ζ7ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ76+ζ7    orthogonal lifted from D7
ρ11202020ζ75+ζ72ζ74+ζ73ζ76+ζ7222ζ74+ζ73ζ75+ζ72ζ76+ζ7000ζ75+ζ72ζ76+ζ7ζ74+ζ73ζ75+ζ72ζ75+ζ72ζ74+ζ73ζ74+ζ73ζ74+ζ73ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ75+ζ72    orthogonal lifted from D7
ρ12202020ζ74+ζ73ζ76+ζ7ζ75+ζ72222ζ76+ζ7ζ74+ζ73ζ75+ζ72000ζ74+ζ73ζ75+ζ72ζ76+ζ7ζ74+ζ73ζ74+ζ73ζ76+ζ7ζ76+ζ7ζ76+ζ7ζ75+ζ72ζ75+ζ72ζ75+ζ72ζ74+ζ73    orthogonal lifted from D7
ρ132200-1-1222ζ98+ζ9ζ97+ζ92ζ95+ζ94000ζ98+ζ9ζ97+ζ92ζ95+ζ94-1-1-1ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ98+ζ9    orthogonal lifted from D9
ρ142200-1-1222ζ97+ζ92ζ95+ζ94ζ98+ζ9000ζ97+ζ92ζ95+ζ94ζ98+ζ9-1-1-1ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ97+ζ92    orthogonal lifted from D9
ρ152-200-11222ζ98+ζ9ζ97+ζ92ζ95+ζ94000-ζ98-ζ9-ζ97-ζ92-ζ95-ζ94-1-1-1ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ98+ζ9    orthogonal lifted from D18
ρ162200-1-1222ζ95+ζ94ζ98+ζ9ζ97+ζ92000ζ95+ζ94ζ98+ζ9ζ97+ζ92-1-1-1ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ95+ζ94    orthogonal lifted from D9
ρ172-200-11222ζ97+ζ92ζ95+ζ94ζ98+ζ9000-ζ97-ζ92-ζ95-ζ94-ζ98-ζ9-1-1-1ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ97+ζ92    orthogonal lifted from D18
ρ182-200-11222ζ95+ζ94ζ98+ζ9ζ97+ζ92000-ζ95-ζ94-ζ98-ζ9-ζ97-ζ92-1-1-1ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ95+ζ94ζ98+ζ9ζ97+ζ92ζ95+ζ94    orthogonal lifted from D18
ρ194000402ζ76+2ζ72ζ75+2ζ722ζ74+2ζ73-2-2-20000002ζ76+2ζ72ζ74+2ζ732ζ75+2ζ72-ζ76-ζ7-ζ76-ζ7-ζ75-ζ72-ζ75-ζ72-ζ75-ζ72-ζ74-ζ73-ζ74-ζ73-ζ74-ζ73-ζ76-ζ7    orthogonal lifted from S3×D7
ρ204000402ζ75+2ζ722ζ74+2ζ732ζ76+2ζ7-2-2-20000002ζ75+2ζ722ζ76+2ζ72ζ74+2ζ73-ζ75-ζ72-ζ75-ζ72-ζ74-ζ73-ζ74-ζ73-ζ74-ζ73-ζ76-ζ7-ζ76-ζ7-ζ76-ζ7-ζ75-ζ72    orthogonal lifted from S3×D7
ρ214000402ζ74+2ζ732ζ76+2ζ72ζ75+2ζ72-2-2-20000002ζ74+2ζ732ζ75+2ζ722ζ76+2ζ7-ζ74-ζ73-ζ74-ζ73-ζ76-ζ7-ζ76-ζ7-ζ76-ζ7-ζ75-ζ72-ζ75-ζ72-ζ75-ζ72-ζ74-ζ73    orthogonal lifted from S3×D7
ρ224000-202ζ76+2ζ72ζ75+2ζ722ζ74+2ζ732ζ97+2ζ922ζ95+2ζ942ζ98+2ζ9000000-ζ76-ζ7-ζ74-ζ73-ζ75-ζ72ζ95ζ76+ζ95ζ7+ζ94ζ76+ζ94ζ7ζ98ζ76+ζ98ζ7+ζ9ζ76+ζ9ζ7ζ97ζ75+ζ97ζ72+ζ92ζ75+ζ92ζ72ζ95ζ75+ζ95ζ72+ζ94ζ75+ζ94ζ72ζ98ζ75+ζ98ζ72+ζ9ζ75+ζ9ζ72ζ97ζ74+ζ97ζ73+ζ92ζ74+ζ92ζ73ζ95ζ74+ζ95ζ73+ζ94ζ74+ζ94ζ73ζ98ζ74+ζ98ζ73+ζ9ζ74+ζ9ζ73ζ97ζ76+ζ97ζ7+ζ92ζ76+ζ92ζ7    orthogonal faithful
ρ234000-202ζ75+2ζ722ζ74+2ζ732ζ76+2ζ72ζ95+2ζ942ζ98+2ζ92ζ97+2ζ92000000-ζ75-ζ72-ζ76-ζ7-ζ74-ζ73ζ98ζ75+ζ98ζ72+ζ9ζ75+ζ9ζ72ζ97ζ75+ζ97ζ72+ζ92ζ75+ζ92ζ72ζ95ζ74+ζ95ζ73+ζ94ζ74+ζ94ζ73ζ98ζ74+ζ98ζ73+ζ9ζ74+ζ9ζ73ζ97ζ74+ζ97ζ73+ζ92ζ74+ζ92ζ73ζ95ζ76+ζ95ζ7+ζ94ζ76+ζ94ζ7ζ98ζ76+ζ98ζ7+ζ9ζ76+ζ9ζ7ζ97ζ76+ζ97ζ7+ζ92ζ76+ζ92ζ7ζ95ζ75+ζ95ζ72+ζ94ζ75+ζ94ζ72    orthogonal faithful
ρ244000-202ζ75+2ζ722ζ74+2ζ732ζ76+2ζ72ζ98+2ζ92ζ97+2ζ922ζ95+2ζ94000000-ζ75-ζ72-ζ76-ζ7-ζ74-ζ73ζ97ζ75+ζ97ζ72+ζ92ζ75+ζ92ζ72ζ95ζ75+ζ95ζ72+ζ94ζ75+ζ94ζ72ζ98ζ74+ζ98ζ73+ζ9ζ74+ζ9ζ73ζ97ζ74+ζ97ζ73+ζ92ζ74+ζ92ζ73ζ95ζ74+ζ95ζ73+ζ94ζ74+ζ94ζ73ζ98ζ76+ζ98ζ7+ζ9ζ76+ζ9ζ7ζ97ζ76+ζ97ζ7+ζ92ζ76+ζ92ζ7ζ95ζ76+ζ95ζ7+ζ94ζ76+ζ94ζ7ζ98ζ75+ζ98ζ72+ζ9ζ75+ζ9ζ72    orthogonal faithful
ρ254000-202ζ76+2ζ72ζ75+2ζ722ζ74+2ζ732ζ98+2ζ92ζ97+2ζ922ζ95+2ζ94000000-ζ76-ζ7-ζ74-ζ73-ζ75-ζ72ζ97ζ76+ζ97ζ7+ζ92ζ76+ζ92ζ7ζ95ζ76+ζ95ζ7+ζ94ζ76+ζ94ζ7ζ98ζ75+ζ98ζ72+ζ9ζ75+ζ9ζ72ζ97ζ75+ζ97ζ72+ζ92ζ75+ζ92ζ72ζ95ζ75+ζ95ζ72+ζ94ζ75+ζ94ζ72ζ98ζ74+ζ98ζ73+ζ9ζ74+ζ9ζ73ζ97ζ74+ζ97ζ73+ζ92ζ74+ζ92ζ73ζ95ζ74+ζ95ζ73+ζ94ζ74+ζ94ζ73ζ98ζ76+ζ98ζ7+ζ9ζ76+ζ9ζ7    orthogonal faithful
ρ264000-202ζ74+2ζ732ζ76+2ζ72ζ75+2ζ722ζ97+2ζ922ζ95+2ζ942ζ98+2ζ9000000-ζ74-ζ73-ζ75-ζ72-ζ76-ζ7ζ95ζ74+ζ95ζ73+ζ94ζ74+ζ94ζ73ζ98ζ74+ζ98ζ73+ζ9ζ74+ζ9ζ73ζ97ζ76+ζ97ζ7+ζ92ζ76+ζ92ζ7ζ95ζ76+ζ95ζ7+ζ94ζ76+ζ94ζ7ζ98ζ76+ζ98ζ7+ζ9ζ76+ζ9ζ7ζ97ζ75+ζ97ζ72+ζ92ζ75+ζ92ζ72ζ95ζ75+ζ95ζ72+ζ94ζ75+ζ94ζ72ζ98ζ75+ζ98ζ72+ζ9ζ75+ζ9ζ72ζ97ζ74+ζ97ζ73+ζ92ζ74+ζ92ζ73    orthogonal faithful
ρ274000-202ζ76+2ζ72ζ75+2ζ722ζ74+2ζ732ζ95+2ζ942ζ98+2ζ92ζ97+2ζ92000000-ζ76-ζ7-ζ74-ζ73-ζ75-ζ72ζ98ζ76+ζ98ζ7+ζ9ζ76+ζ9ζ7ζ97ζ76+ζ97ζ7+ζ92ζ76+ζ92ζ7ζ95ζ75+ζ95ζ72+ζ94ζ75+ζ94ζ72ζ98ζ75+ζ98ζ72+ζ9ζ75+ζ9ζ72ζ97ζ75+ζ97ζ72+ζ92ζ75+ζ92ζ72ζ95ζ74+ζ95ζ73+ζ94ζ74+ζ94ζ73ζ98ζ74+ζ98ζ73+ζ9ζ74+ζ9ζ73ζ97ζ74+ζ97ζ73+ζ92ζ74+ζ92ζ73ζ95ζ76+ζ95ζ7+ζ94ζ76+ζ94ζ7    orthogonal faithful
ρ284000-202ζ74+2ζ732ζ76+2ζ72ζ75+2ζ722ζ98+2ζ92ζ97+2ζ922ζ95+2ζ94000000-ζ74-ζ73-ζ75-ζ72-ζ76-ζ7ζ97ζ74+ζ97ζ73+ζ92ζ74+ζ92ζ73ζ95ζ74+ζ95ζ73+ζ94ζ74+ζ94ζ73ζ98ζ76+ζ98ζ7+ζ9ζ76+ζ9ζ7ζ97ζ76+ζ97ζ7+ζ92ζ76+ζ92ζ7ζ95ζ76+ζ95ζ7+ζ94ζ76+ζ94ζ7ζ98ζ75+ζ98ζ72+ζ9ζ75+ζ9ζ72ζ97ζ75+ζ97ζ72+ζ92ζ75+ζ92ζ72ζ95ζ75+ζ95ζ72+ζ94ζ75+ζ94ζ72ζ98ζ74+ζ98ζ73+ζ9ζ74+ζ9ζ73    orthogonal faithful
ρ294000-202ζ74+2ζ732ζ76+2ζ72ζ75+2ζ722ζ95+2ζ942ζ98+2ζ92ζ97+2ζ92000000-ζ74-ζ73-ζ75-ζ72-ζ76-ζ7ζ98ζ74+ζ98ζ73+ζ9ζ74+ζ9ζ73ζ97ζ74+ζ97ζ73+ζ92ζ74+ζ92ζ73ζ95ζ76+ζ95ζ7+ζ94ζ76+ζ94ζ7ζ98ζ76+ζ98ζ7+ζ9ζ76+ζ9ζ7ζ97ζ76+ζ97ζ7+ζ92ζ76+ζ92ζ7ζ95ζ75+ζ95ζ72+ζ94ζ75+ζ94ζ72ζ98ζ75+ζ98ζ72+ζ9ζ75+ζ9ζ72ζ97ζ75+ζ97ζ72+ζ92ζ75+ζ92ζ72ζ95ζ74+ζ95ζ73+ζ94ζ74+ζ94ζ73    orthogonal faithful
ρ304000-202ζ75+2ζ722ζ74+2ζ732ζ76+2ζ72ζ97+2ζ922ζ95+2ζ942ζ98+2ζ9000000-ζ75-ζ72-ζ76-ζ7-ζ74-ζ73ζ95ζ75+ζ95ζ72+ζ94ζ75+ζ94ζ72ζ98ζ75+ζ98ζ72+ζ9ζ75+ζ9ζ72ζ97ζ74+ζ97ζ73+ζ92ζ74+ζ92ζ73ζ95ζ74+ζ95ζ73+ζ94ζ74+ζ94ζ73ζ98ζ74+ζ98ζ73+ζ9ζ74+ζ9ζ73ζ97ζ76+ζ97ζ7+ζ92ζ76+ζ92ζ7ζ95ζ76+ζ95ζ7+ζ94ζ76+ζ94ζ7ζ98ζ76+ζ98ζ7+ζ9ζ76+ζ9ζ7ζ97ζ75+ζ97ζ72+ζ92ζ75+ζ92ζ72    orthogonal faithful

Smallest permutation representation of D7×D9
►On 63 points
Generators in S63
(1 40 15 62 19 52 28)(2 41 16 63 20 53 29)(3 42 17 55 21 54 30)(4 43 18 56 22 46 31)(5 44 10 57 23 47 32)(6 45 11 58 24 48 33)(7 37 12 59 25 49 34)(8 38 13 60 26 50 35)(9 39 14 61 27 51 36)

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

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

Matrix representation of D7×D9 ►in GL4(𝔽127) generated by

90100
286100
0010
0001
,
612400
996600
0010
0001
,
1000
0100
00922
0010531
,
1000
0100
0010531
00922
G:=sub<GL(4,GF(127))| [90,28,0,0,1,61,0,0,0,0,1,0,0,0,0,1],[61,99,0,0,24,66,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,9,105,0,0,22,31],[1,0,0,0,0,1,0,0,0,0,105,9,0,0,31,22] >;
 

D7×D9 in GAP, Magma, Sage, TeX 

D_7\times D_9
 
% in TeX
 
G:=Group("D7xD9");
 
// GroupNames label
 
G:=SmallGroup(252,8);
 
// by ID
 
G=gap.SmallGroup(252,8);
 
# by ID
 
G:=PCGroup([5,-2,-2,-3,-7,-3,697,642,1443,2109]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^7=b^2=c^9=d^2=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=c^-1>;
 
// generators/relations
 

Export

Subgroup lattice of D7×D9 in TeX
Character table of D7×D9 in TeX

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