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G = C35⋊C12order 420 = 22·3·5·7

1st semidirect product of C35 and C12 acting faithfully

metacyclic, supersoluble, monomial, Z-group

Aliases: C351C12, D5.F7, C7⋊F5⋊C3, C7⋊C3⋊F5, C7⋊(C3×F5), C5⋊(C7⋊C12), (C7×D5).1C6, (C5×C7⋊C3)⋊1C4, (D5×C7⋊C3).1C2, SmallGroup(420,15)

Series: Derived Chief Lower central Upper central

C1C35 — C35⋊C12
C1C7C35C7×D5D5×C7⋊C3 — C35⋊C12
C35 — C35⋊C12
C1

Generators and relations for C35⋊C12
 G = < a,b | a35=b12=1, bab-1=a17 >

5C2
7C3
35C4
35C6
5C14
7C15
35C12
7F5
5Dic7
7C3×D5
5C2×C7⋊C3
7C3×F5
5C7⋊C12

Character table of C35⋊C12

 class 123A3B4A4B56A6B712A12B12C12D1415A15B35A35B
 size 15773535435356353535353028281212
ρ11111111111111111111    trivial
ρ21111-1-11111-1-1-1-111111    linear of order 2
ρ311ζ3ζ32111ζ3ζ321ζ32ζ3ζ3ζ321ζ3ζ3211    linear of order 3
ρ411ζ32ζ3-1-11ζ32ζ31ζ65ζ6ζ6ζ651ζ32ζ311    linear of order 6
ρ511ζ32ζ3111ζ32ζ31ζ3ζ32ζ32ζ31ζ32ζ311    linear of order 3
ρ611ζ3ζ32-1-11ζ3ζ321ζ6ζ65ζ65ζ61ζ3ζ3211    linear of order 6
ρ71-111i-i1-1-11i-ii-i-11111    linear of order 4
ρ81-111-ii1-1-11-ii-ii-11111    linear of order 4
ρ91-1ζ3ζ32i-i1ζ65ζ61ζ4ζ32ζ43ζ3ζ4ζ3ζ43ζ32-1ζ3ζ3211    linear of order 12
ρ101-1ζ32ζ3i-i1ζ6ζ651ζ4ζ3ζ43ζ32ζ4ζ32ζ43ζ3-1ζ32ζ311    linear of order 12
ρ111-1ζ3ζ32-ii1ζ65ζ61ζ43ζ32ζ4ζ3ζ43ζ3ζ4ζ32-1ζ3ζ3211    linear of order 12
ρ121-1ζ32ζ3-ii1ζ6ζ651ζ43ζ3ζ4ζ32ζ43ζ32ζ4ζ3-1ζ32ζ311    linear of order 12
ρ13404400-100400000-1-1-1-1    orthogonal lifted from F5
ρ1440-2+2-3-2-2-300-100400000ζ65ζ6-1-1    complex lifted from C3×F5
ρ1540-2-2-3-2+2-300-100400000ζ6ζ65-1-1    complex lifted from C3×F5
ρ16660000600-10000-100-1-1    orthogonal lifted from F7
ρ176-60000600-10000100-1-1    symplectic lifted from C7⋊C12, Schur index 2
ρ181200000-300-200000001+-35/21--35/2    complex faithful
ρ191200000-300-200000001--35/21+-35/2    complex faithful

Smallest permutation representation of C35⋊C12
On 35 points
Generators in S35
(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 34 5 28 17 4 30 13 12 14 10 18)(3 32 9 20 33 7 24 25 23 27 19 35)(6 26 21 31 11 16)(8 22 29 15)

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

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), (2,34,5,28,17,4,30,13,12,14,10,18)(3,32,9,20,33,7,24,25,23,27,19,35)(6,26,21,31,11,16)(8,22,29,15) );

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)], [(2,34,5,28,17,4,30,13,12,14,10,18),(3,32,9,20,33,7,24,25,23,27,19,35),(6,26,21,31,11,16),(8,22,29,15)]])

Matrix representation of C35⋊C12 in GL10(ℤ)

0100000000
0010000000
0001000000
-1-1-1-1000000
000000-1100
000000-1010
000000-1001
000000-1000
000010-1000
000001-1000
,
1000000000
0010000000
-1-1-1-1000000
0100000000
0000001000
0000000001
0000010000
0000000010
0000100000
0000000100

G:=sub<GL(10,Integers())| [0,0,0,-1,0,0,0,0,0,0,1,0,0,-1,0,0,0,0,0,0,0,1,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,-1,-1,-1,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0],[1,0,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0] >;

C35⋊C12 in GAP, Magma, Sage, TeX

C_{35}\rtimes C_{12}
% in TeX

G:=Group("C35:C12");
// GroupNames label

G:=SmallGroup(420,15);
// by ID

G=gap.SmallGroup(420,15);
# by ID

G:=PCGroup([5,-2,-3,-2,-5,-7,30,723,173,9004,1509]);
// Polycyclic

G:=Group<a,b|a^35=b^12=1,b*a*b^-1=a^17>;
// generators/relations

Export

Subgroup lattice of C35⋊C12 in TeX
Character table of C35⋊C12 in TeX

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