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G = D5×F7  order 420 = 22·3·5·7

Direct product of D5 and F7

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

Aliases: D5×F7, D35⋊C6, C5⋊F7⋊C2, C35⋊(C2×C6), (D5×D7)⋊C3, D7⋊(C3×D5), (C5×D7)⋊C6, (C7×D5)⋊C6, C7⋊C3⋊1D10, C7⋊1(C6×D5), (C5×F7)⋊C2, C5⋊1(C2×F7), (D5×C7⋊C3)⋊C2, (C5×C7⋊C3)⋊C22, SmallGroup(420,16)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C35 — D5×F7
Chief series
C1 — C7 — C35 — C5×C7⋊C3 — C5×F7 — D5×F7
Lower central
C35 — D5×F7
Upper central
C1

Generators and relations for D5×F7
 G = < a,b,c,d | a5=b2=c7=d6=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >

C1
5C2
7C2
35C2
7C3
C5
C7
35C22
7C6
35C6
35C6
D5
7C10
7D5
D7
5D7
5C14
7C15
C7⋊C3
C35
35C2×C6
7D10
5D14
7C3×D5
7C3×D5
7C30
F7
5F7
5C2×C7⋊C3
C5×D7
C7×D5
D35
C5×C7⋊C3
7C6×D5
5C2×F7
D5×D7
C5×F7
C5⋊F7
D5×C7⋊C3
D5×F7

Character table of D5×F7

 class 12A2B2C3A3B5A5B6A6B6C6D6E6F710A10B1415A15B15C15D30A30B30C30D35A35B
 size 1573577227735353535614143014141414141414141212
ρ11111111111111111111111111111    trivial
ρ21-1-111111-1-11-11-11-1-1-11111-1-1-1-111    linear of order 2
ρ311-1-11111-1-1-11-111-1-111111-1-1-1-111    linear of order 2
ρ41-11-1111111-1-1-1-1111-11111111111    linear of order 2
ρ51-11-1ζ3ζ3211ζ3ζ32ζ65ζ65ζ6ζ6111-1ζ32ζ32ζ3ζ3ζ32ζ3ζ32ζ311    linear of order 6
ρ61111ζ32ζ311ζ32ζ3ζ32ζ32ζ3ζ31111ζ3ζ3ζ32ζ32ζ3ζ32ζ3ζ3211    linear of order 3
ρ711-1-1ζ32ζ311ζ6ζ65ζ6ζ32ζ65ζ31-1-11ζ3ζ3ζ32ζ32ζ65ζ6ζ65ζ611    linear of order 6
ρ81111ζ3ζ3211ζ3ζ32ζ3ζ3ζ32ζ321111ζ32ζ32ζ3ζ3ζ32ζ3ζ32ζ311    linear of order 3
ρ91-1-11ζ3ζ3211ζ65ζ6ζ3ζ65ζ32ζ61-1-1-1ζ32ζ32ζ3ζ3ζ6ζ65ζ6ζ6511    linear of order 6
ρ1011-1-1ζ3ζ3211ζ65ζ6ζ65ζ3ζ6ζ321-1-11ζ32ζ32ζ3ζ3ζ6ζ65ζ6ζ6511    linear of order 6
ρ111-11-1ζ32ζ311ζ32ζ3ζ6ζ6ζ65ζ65111-1ζ3ζ3ζ32ζ32ζ3ζ32ζ3ζ3211    linear of order 6
ρ121-1-11ζ32ζ311ζ6ζ65ζ32ζ6ζ3ζ651-1-1-1ζ3ζ3ζ32ζ32ζ65ζ6ζ65ζ611    linear of order 6
ρ1320-2022-1-√5/2-1+√5/2-2-2000021-√5/21+√5/20-1-√5/2-1+√5/2-1+√5/2-1-√5/21+√5/21+√5/21-√5/21-√5/2-1-√5/2-1+√5/2    orthogonal lifted from D10
ρ1420-2022-1+√5/2-1-√5/2-2-2000021+√5/21-√5/20-1+√5/2-1-√5/2-1-√5/2-1+√5/21-√5/21-√5/21+√5/21+√5/2-1+√5/2-1-√5/2    orthogonal lifted from D10
ρ15202022-1-√5/2-1+√5/22200002-1+√5/2-1-√5/20-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/2-1+√5/2    orthogonal lifted from D5
ρ16202022-1+√5/2-1-√5/22200002-1-√5/2-1+√5/20-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1-√5/2    orthogonal lifted from D5
ρ1720-20-1+√-3-1-√-3-1+√5/2-1-√5/21-√-31+√-3000021+√5/21-√5/20ζ32ζ54+ζ32ζ5ζ32ζ53+ζ32ζ52ζ3ζ53+ζ3ζ52ζ3ζ54+ζ3ζ5-ζ32ζ54-ζ32ζ5-ζ3ζ54-ζ3ζ5-ζ32ζ53-ζ32ζ52-ζ3ζ53-ζ3ζ52-1+√5/2-1-√5/2    complex lifted from C6×D5
ρ1820-20-1+√-3-1-√-3-1-√5/2-1+√5/21-√-31+√-3000021-√5/21+√5/20ζ32ζ53+ζ32ζ52ζ32ζ54+ζ32ζ5ζ3ζ54+ζ3ζ5ζ3ζ53+ζ3ζ52-ζ32ζ53-ζ32ζ52-ζ3ζ53-ζ3ζ52-ζ32ζ54-ζ32ζ5-ζ3ζ54-ζ3ζ5-1-√5/2-1+√5/2    complex lifted from C6×D5
ρ192020-1-√-3-1+√-3-1+√5/2-1-√5/2-1-√-3-1+√-300002-1-√5/2-1+√5/20ζ3ζ54+ζ3ζ5ζ3ζ53+ζ3ζ52ζ32ζ53+ζ32ζ52ζ32ζ54+ζ32ζ5ζ3ζ54+ζ3ζ5ζ32ζ54+ζ32ζ5ζ3ζ53+ζ3ζ52ζ32ζ53+ζ32ζ52-1+√5/2-1-√5/2    complex lifted from C3×D5
ρ2020-20-1-√-3-1+√-3-1+√5/2-1-√5/21+√-31-√-3000021+√5/21-√5/20ζ3ζ54+ζ3ζ5ζ3ζ53+ζ3ζ52ζ32ζ53+ζ32ζ52ζ32ζ54+ζ32ζ5-ζ3ζ54-ζ3ζ5-ζ32ζ54-ζ32ζ5-ζ3ζ53-ζ3ζ52-ζ32ζ53-ζ32ζ52-1+√5/2-1-√5/2    complex lifted from C6×D5
ρ212020-1+√-3-1-√-3-1-√5/2-1+√5/2-1+√-3-1-√-300002-1+√5/2-1-√5/20ζ32ζ53+ζ32ζ52ζ32ζ54+ζ32ζ5ζ3ζ54+ζ3ζ5ζ3ζ53+ζ3ζ52ζ32ζ53+ζ32ζ52ζ3ζ53+ζ3ζ52ζ32ζ54+ζ32ζ5ζ3ζ54+ζ3ζ5-1-√5/2-1+√5/2    complex lifted from C3×D5
ρ222020-1-√-3-1+√-3-1-√5/2-1+√5/2-1-√-3-1+√-300002-1+√5/2-1-√5/20ζ3ζ53+ζ3ζ52ζ3ζ54+ζ3ζ5ζ32ζ54+ζ32ζ5ζ32ζ53+ζ32ζ52ζ3ζ53+ζ3ζ52ζ32ζ53+ζ32ζ52ζ3ζ54+ζ3ζ5ζ32ζ54+ζ32ζ5-1-√5/2-1+√5/2    complex lifted from C3×D5
ρ232020-1+√-3-1-√-3-1+√5/2-1-√5/2-1+√-3-1-√-300002-1-√5/2-1+√5/20ζ32ζ54+ζ32ζ5ζ32ζ53+ζ32ζ52ζ3ζ53+ζ3ζ52ζ3ζ54+ζ3ζ5ζ32ζ54+ζ32ζ5ζ3ζ54+ζ3ζ5ζ32ζ53+ζ32ζ52ζ3ζ53+ζ3ζ52-1+√5/2-1-√5/2    complex lifted from C3×D5
ρ2420-20-1-√-3-1+√-3-1-√5/2-1+√5/21+√-31-√-3000021-√5/21+√5/20ζ3ζ53+ζ3ζ52ζ3ζ54+ζ3ζ5ζ32ζ54+ζ32ζ5ζ32ζ53+ζ32ζ52-ζ3ζ53-ζ3ζ52-ζ32ζ53-ζ32ζ52-ζ3ζ54-ζ3ζ5-ζ32ζ54-ζ32ζ5-1-√5/2-1+√5/2    complex lifted from C6×D5
ρ256-6000066000000-100100000000-1-1    orthogonal lifted from C2×F7
ρ2666000066000000-100-100000000-1-1    orthogonal lifted from F7
ρ271200000-3+3√5-3-3√5000000-2000000000001-√5/21+√5/2    orthogonal faithful
ρ281200000-3-3√5-3+3√5000000-2000000000001+√5/21-√5/2    orthogonal faithful

Smallest permutation representation of D5×F7
►On 35 points
Generators in S35
(1 29 22 15 8)(2 30 23 16 9)(3 31 24 17 10)(4 32 25 18 11)(5 33 26 19 12)(6 34 27 20 13)(7 35 28 21 14)

(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(21 35)

(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)

(2 4 3 7 5 6)(9 11 10 14 12 13)(16 18 17 21 19 20)(23 25 24 28 26 27)(30 32 31 35 33 34)
 
G:=sub<Sym(35)| (1,29,22,15,8)(2,30,23,16,9)(3,31,24,17,10)(4,32,25,18,11)(5,33,26,19,12)(6,34,27,20,13)(7,35,28,21,14), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35), (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), (2,4,3,7,5,6)(9,11,10,14,12,13)(16,18,17,21,19,20)(23,25,24,28,26,27)(30,32,31,35,33,34)>;
 
G:=Group( (1,29,22,15,8)(2,30,23,16,9)(3,31,24,17,10)(4,32,25,18,11)(5,33,26,19,12)(6,34,27,20,13)(7,35,28,21,14), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35), (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), (2,4,3,7,5,6)(9,11,10,14,12,13)(16,18,17,21,19,20)(23,25,24,28,26,27)(30,32,31,35,33,34) );
 
G=PermutationGroup([[(1,29,22,15,8),(2,30,23,16,9),(3,31,24,17,10),(4,32,25,18,11),(5,33,26,19,12),(6,34,27,20,13),(7,35,28,21,14)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,29),(16,30),(17,31),(18,32),(19,33),(20,34),(21,35)], [(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)], [(2,4,3,7,5,6),(9,11,10,14,12,13),(16,18,17,21,19,20),(23,25,24,28,26,27),(30,32,31,35,33,34)]])
 

Matrix representation of D5×F7 ►in GL8(𝔽211)

2101000000
31179000000
00100000
00010000
00001000
00000100
00000010
00000001
,
10000000
180210000000
00100000
00010000
00001000
00000100
00000010
00000001
,
10000000
01000000
00210210210210210210
00100000
00010000
00001000
00000100
00000010
,
150000000
015000000
00100000
00000001
00000100
00010000
00210210210210210210
00000010

G:=sub<GL(8,GF(211))| [210,31,0,0,0,0,0,0,1,179,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,180,0,0,0,0,0,0,0,210,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,210,1,0,0,0,0,0,0,210,0,1,0,0,0,0,0,210,0,0,1,0,0,0,0,210,0,0,0,1,0,0,0,210,0,0,0,0,1,0,0,210,0,0,0,0,0],[15,0,0,0,0,0,0,0,0,15,0,0,0,0,0,0,0,0,1,0,0,0,210,0,0,0,0,0,0,1,210,0,0,0,0,0,0,0,210,0,0,0,0,0,1,0,210,0,0,0,0,0,0,0,210,1,0,0,0,1,0,0,210,0] >;
 

D5×F7 in GAP, Magma, Sage, TeX 

D_5\times F_7
 
% in TeX
 
G:=Group("D5xF7");
 
// GroupNames label
 
G:=SmallGroup(420,16);
 
// by ID
 
G=gap.SmallGroup(420,16);
 
# by ID
 
G:=PCGroup([5,-2,-2,-3,-5,-7,488,9004,1514]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^5=b^2=c^7=d^6=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^5>;
 
// generators/relations
 

Export

Subgroup lattice of D5×F7 in TeX
Character table of D5×F7 in TeX

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