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G = D26⋊C6order 312 = 23·3·13

2nd semidirect product of D26 and C6 acting faithfully

metabelian, supersoluble, monomial

Aliases: D262C6, Dic13⋊C6, C13⋊D4⋊C3, C13⋊C32D4, C26.C6⋊C2, C132(C3×D4), (C2×C26)⋊3C6, C26.5(C2×C6), C222(C13⋊C6), (C2×C13⋊C6)⋊2C2, C2.5(C2×C13⋊C6), (C22×C13⋊C3)⋊1C2, (C2×C13⋊C3).5C22, SmallGroup(312,12)

Series: Derived Chief Lower central Upper central

C1C26 — D26⋊C6
C1C13C26C2×C13⋊C3C2×C13⋊C6 — D26⋊C6
C13C26 — D26⋊C6
C1C2C22

Generators and relations for D26⋊C6
 G = < a,b,c | a26=b2=c6=1, bab=a-1, cac-1=a3, cbc-1=a15b >

2C2
26C2
13C3
13C22
13C4
13C6
26C6
26C6
2D13
2C26
13D4
13C2×C6
13C2×C6
13C12
2C2×C13⋊C3
2C13⋊C6
13C3×D4

Character table of D26⋊C6

 class 12A2B2C3A3B46A6B6C6D6E6F12A12B13A13B26A26B26C26D26E26F
 size 11226131326131326262626262666666666
ρ111111111111111111111111    trivial
ρ2111-111-1111-1-11-1-111111111    linear of order 2
ρ311-1-111111-1-1-1-11111-11-1-1-11    linear of order 2
ρ411-1111-111-111-1-1-111-11-1-1-11    linear of order 2
ρ511-11ζ32ζ3-1ζ32ζ3ζ6ζ32ζ3ζ65ζ65ζ611-11-1-1-11    linear of order 6
ρ61111ζ3ζ321ζ3ζ32ζ3ζ3ζ32ζ32ζ32ζ311111111    linear of order 3
ρ711-11ζ3ζ32-1ζ3ζ32ζ65ζ3ζ32ζ6ζ6ζ6511-11-1-1-11    linear of order 6
ρ8111-1ζ32ζ3-1ζ32ζ3ζ32ζ6ζ65ζ3ζ65ζ611111111    linear of order 6
ρ91111ζ32ζ31ζ32ζ3ζ32ζ32ζ3ζ3ζ3ζ3211111111    linear of order 3
ρ1011-1-1ζ3ζ321ζ3ζ32ζ65ζ65ζ6ζ6ζ32ζ311-11-1-1-11    linear of order 6
ρ11111-1ζ3ζ32-1ζ3ζ32ζ3ζ65ζ6ζ32ζ6ζ6511111111    linear of order 6
ρ1211-1-1ζ32ζ31ζ32ζ3ζ6ζ6ζ65ζ65ζ3ζ3211-11-1-1-11    linear of order 6
ρ132-200220-2-2000000220-2000-2    orthogonal lifted from D4
ρ142-200-1--3-1+-301+-31--3000000220-2000-2    complex lifted from C3×D4
ρ152-200-1+-3-1--301--31+-3000000220-2000-2    complex lifted from C3×D4
ρ16666000000000000-1+13/2-1-13/2-1-13/2-1+13/2-1+13/2-1-13/2-1+13/2-1-13/2    orthogonal lifted from C13⋊C6
ρ17666000000000000-1-13/2-1+13/2-1+13/2-1-13/2-1-13/2-1+13/2-1-13/2-1+13/2    orthogonal lifted from C13⋊C6
ρ1866-6000000000000-1+13/2-1-13/21+13/2-1+13/21-13/21+13/21-13/2-1-13/2    orthogonal lifted from C2×C13⋊C6
ρ1966-6000000000000-1-13/2-1+13/21-13/2-1-13/21+13/21-13/21+13/2-1+13/2    orthogonal lifted from C2×C13⋊C6
ρ206-60000000000000-1+13/2-1-13/213111381371361351321-13/2ζ1312131013913413313ζ131113813713613513213121310139134133131+13/2    complex faithful
ρ216-60000000000000-1-13/2-1+13/213121310139134133131+13/21311138137136135132ζ1312131013913413313ζ13111381371361351321-13/2    complex faithful
ρ226-60000000000000-1+13/2-1-13/2ζ13111381371361351321-13/213121310139134133131311138137136135132ζ13121310139134133131+13/2    complex faithful
ρ236-60000000000000-1-13/2-1+13/2ζ13121310139134133131+13/2ζ1311138137136135132131213101391341331313111381371361351321-13/2    complex faithful

Smallest permutation representation of D26⋊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 41)(2 40)(3 39)(4 38)(5 37)(6 36)(7 35)(8 34)(9 33)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 52)(17 51)(18 50)(19 49)(20 48)(21 47)(22 46)(23 45)(24 44)(25 43)(26 42)
(2 10 4)(3 19 7)(5 11 13)(6 20 16)(8 12 22)(9 21 25)(15 23 17)(18 24 26)(27 50 49 40 37 36)(28 33 52 41 46 39)(29 42)(30 51 32 43 38 45)(31 34 35 44 47 48)

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

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

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

Matrix representation of D26⋊C6 in GL6(𝔽3)

200001
001020
011020
100100
001000
000200
,
010000
100000
000001
000020
000200
001000
,
200002
021000
012010
100201
002020
100102

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

D26⋊C6 in GAP, Magma, Sage, TeX

D_{26}\rtimes C_6
% in TeX

G:=Group("D26:C6");
// GroupNames label

G:=SmallGroup(312,12);
// by ID

G=gap.SmallGroup(312,12);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-13,141,7204,464]);
// Polycyclic

G:=Group<a,b,c|a^26=b^2=c^6=1,b*a*b=a^-1,c*a*c^-1=a^3,c*b*c^-1=a^15*b>;
// generators/relations

Export

Subgroup lattice of D26⋊C6 in TeX
Character table of D26⋊C6 in TeX

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