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G = C26.C6order 156 = 22·3·13

The non-split extension by C26 of C6 acting faithfully

metacyclic, supersoluble, monomial, Z-group

Aliases: C26.C6, C132C12, Dic13⋊C3, C13⋊C32C4, C2.(C13⋊C6), (C2×C13⋊C3).C2, SmallGroup(156,1)

Series: Derived Chief Lower central Upper central

C1C13 — C26.C6
C1C13C26C2×C13⋊C3 — C26.C6
C13 — C26.C6
C1C2

Generators and relations for C26.C6
 G = < a,b | a26=1, b6=a13, bab-1=a23 >

13C3
13C4
13C6
13C12

Character table of C26.C6

 class 123A3B4A4B6A6B12A12B12C12D13A13B26A26B
 size 11131313131313131313136666
ρ11111111111111111    trivial
ρ21111-1-111-1-1-1-11111    linear of order 2
ρ311ζ3ζ3211ζ3ζ32ζ3ζ32ζ32ζ31111    linear of order 3
ρ411ζ32ζ3-1-1ζ32ζ3ζ6ζ65ζ65ζ61111    linear of order 6
ρ511ζ32ζ311ζ32ζ3ζ32ζ3ζ3ζ321111    linear of order 3
ρ611ζ3ζ32-1-1ζ3ζ32ζ65ζ6ζ6ζ651111    linear of order 6
ρ71-111i-i-1-1i-ii-i11-1-1    linear of order 4
ρ81-111-ii-1-1-ii-ii11-1-1    linear of order 4
ρ91-1ζ3ζ32-iiζ65ζ6ζ43ζ3ζ4ζ32ζ43ζ32ζ4ζ311-1-1    linear of order 12
ρ101-1ζ32ζ3i-iζ6ζ65ζ4ζ32ζ43ζ3ζ4ζ3ζ43ζ3211-1-1    linear of order 12
ρ111-1ζ32ζ3-iiζ6ζ65ζ43ζ32ζ4ζ3ζ43ζ3ζ4ζ3211-1-1    linear of order 12
ρ121-1ζ3ζ32i-iζ65ζ6ζ4ζ3ζ43ζ32ζ4ζ32ζ43ζ311-1-1    linear of order 12
ρ13660000000000-1-13/2-1+13/2-1-13/2-1+13/2    orthogonal lifted from C13⋊C6
ρ14660000000000-1+13/2-1-13/2-1+13/2-1-13/2    orthogonal lifted from C13⋊C6
ρ156-60000000000-1+13/2-1-13/21-13/21+13/2    symplectic faithful, Schur index 2
ρ166-60000000000-1-13/2-1+13/21+13/21-13/2    symplectic faithful, Schur index 2

Smallest permutation representation of C26.C6
On 52 points
Generators in S52
(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)
(1 31 14 44)(2 48 17 43 10 28 15 35 4 30 23 41)(3 39 20 42 19 51 16 52 7 29 6 38)(5 47 26 40 11 45 18 34 13 27 24 32)(8 46 9 37 12 36 21 33 22 50 25 49)

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

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

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

C26.C6 is a maximal subgroup of   C13⋊C24  Dic26⋊C3  C4×C13⋊C6  D26⋊C6  C393C12
C26.C6 is a maximal quotient of   C132C24  C132C36  C393C12

Matrix representation of C26.C6 in GL6(𝔽3)

010000
110001
002200
002120
000210
120002
,
002010
000220
110000
000002
020000
000010

G:=sub<GL(6,GF(3))| [0,1,0,0,0,1,1,1,0,0,0,2,0,0,2,2,0,0,0,0,2,1,2,0,0,0,0,2,1,0,0,1,0,0,0,2],[0,0,1,0,0,0,0,0,1,0,2,0,2,0,0,0,0,0,0,2,0,0,0,0,1,2,0,0,0,1,0,0,0,2,0,0] >;

C26.C6 in GAP, Magma, Sage, TeX

C_{26}.C_6
% in TeX

G:=Group("C26.C6");
// GroupNames label

G:=SmallGroup(156,1);
// by ID

G=gap.SmallGroup(156,1);
# by ID

G:=PCGroup([4,-2,-3,-2,-13,24,2307,295]);
// Polycyclic

G:=Group<a,b|a^26=1,b^6=a^13,b*a*b^-1=a^23>;
// generators/relations

Export

Subgroup lattice of C26.C6 in TeX
Character table of C26.C6 in TeX

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