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G = Dic26⋊C3order 312 = 23·3·13

The semidirect product of Dic26 and C3 acting faithfully

metacyclic, supersoluble, monomial

Aliases: Dic26⋊C3, C52.1C6, Dic13.2C6, C13⋊C3⋊Q8, C13⋊(C3×Q8), C4.(C13⋊C6), C26.1(C2×C6), C26.C6.2C2, C2.3(C2×C13⋊C6), (C4×C13⋊C3).1C2, (C2×C13⋊C3).1C22, SmallGroup(312,8)

Series: Derived Chief Lower central Upper central

C1C26 — Dic26⋊C3
C1C13C26C2×C13⋊C3C26.C6 — Dic26⋊C3
C13C26 — Dic26⋊C3
C1C2C4

Generators and relations for Dic26⋊C3
 G = < a,b,c | a52=c3=1, b2=a26, bab-1=a-1, cac-1=a9, bc=cb >

13C3
13C4
13C4
13C6
13Q8
13C12
13C12
13C12
13C3×Q8

Character table of Dic26⋊C3

 class 123A3B4A4B4C6A6B12A12B12C12D12E12F13A13B26A26B52A52B52C52D
 size 11131322626131326262626262666666666
ρ111111111111111111111111    trivial
ρ211111-1-111-1-1-11-1111111111    linear of order 2
ρ31111-1-11111-1-1-11-11111-1-1-1-1    linear of order 2
ρ41111-11-111-111-1-1-11111-1-1-1-1    linear of order 2
ρ511ζ3ζ321-1-1ζ3ζ32ζ65ζ6ζ65ζ3ζ6ζ3211111111    linear of order 6
ρ611ζ3ζ32-11-1ζ3ζ32ζ65ζ32ζ3ζ65ζ6ζ61111-1-1-1-1    linear of order 6
ρ711ζ3ζ32-1-11ζ3ζ32ζ3ζ6ζ65ζ65ζ32ζ61111-1-1-1-1    linear of order 6
ρ811ζ32ζ3-1-11ζ32ζ3ζ32ζ65ζ6ζ6ζ3ζ651111-1-1-1-1    linear of order 6
ρ911ζ32ζ31-1-1ζ32ζ3ζ6ζ65ζ6ζ32ζ65ζ311111111    linear of order 6
ρ1011ζ32ζ3111ζ32ζ3ζ32ζ3ζ32ζ32ζ3ζ311111111    linear of order 3
ρ1111ζ32ζ3-11-1ζ32ζ3ζ6ζ3ζ32ζ6ζ65ζ651111-1-1-1-1    linear of order 6
ρ1211ζ3ζ32111ζ3ζ32ζ3ζ32ζ3ζ3ζ32ζ3211111111    linear of order 3
ρ132-222000-2-200000022-2-20000    symplectic lifted from Q8, Schur index 2
ρ142-2-1--3-1+-30001+-31--300000022-2-20000    complex lifted from C3×Q8
ρ152-2-1+-3-1--30001--31+-300000022-2-20000    complex lifted from C3×Q8
ρ166600-60000000000-1-13/2-1+13/2-1+13/2-1-13/21-13/21-13/21+13/21+13/2    orthogonal lifted from C2×C13⋊C6
ρ17660060000000000-1+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
ρ186600-60000000000-1+13/2-1-13/2-1-13/2-1+13/21+13/21+13/21-13/21-13/2    orthogonal lifted from C2×C13⋊C6
ρ19660060000000000-1-13/2-1+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
ρ206-60000000000000-1-13/2-1+13/21-13/21+13/2ζ43ζ131243ζ131043ζ13943ζ13443ζ13343ζ13ζ4ζ13124ζ13104ζ1394ζ1344ζ1334ζ1343ζ131143ζ13843ζ13743ζ13643ζ13543ζ1324ζ13114ζ1384ζ1374ζ1364ζ1354ζ132    symplectic faithful, Schur index 2
ρ216-60000000000000-1+13/2-1-13/21+13/21-13/243ζ131143ζ13843ζ13743ζ13643ζ13543ζ1324ζ13114ζ1384ζ1374ζ1364ζ1354ζ132ζ4ζ13124ζ13104ζ1394ζ1344ζ1334ζ13ζ43ζ131243ζ131043ζ13943ζ13443ζ13343ζ13    symplectic faithful, Schur index 2
ρ226-60000000000000-1+13/2-1-13/21+13/21-13/24ζ13114ζ1384ζ1374ζ1364ζ1354ζ13243ζ131143ζ13843ζ13743ζ13643ζ13543ζ132ζ43ζ131243ζ131043ζ13943ζ13443ζ13343ζ13ζ4ζ13124ζ13104ζ1394ζ1344ζ1334ζ13    symplectic faithful, Schur index 2
ρ236-60000000000000-1-13/2-1+13/21-13/21+13/2ζ4ζ13124ζ13104ζ1394ζ1344ζ1334ζ13ζ43ζ131243ζ131043ζ13943ζ13443ζ13343ζ134ζ13114ζ1384ζ1374ζ1364ζ1354ζ13243ζ131143ζ13843ζ13743ζ13643ζ13543ζ132    symplectic faithful, Schur index 2

Smallest permutation representation of Dic26⋊C3
On 104 points
Generators in S104
(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 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104)
(1 92 27 66)(2 91 28 65)(3 90 29 64)(4 89 30 63)(5 88 31 62)(6 87 32 61)(7 86 33 60)(8 85 34 59)(9 84 35 58)(10 83 36 57)(11 82 37 56)(12 81 38 55)(13 80 39 54)(14 79 40 53)(15 78 41 104)(16 77 42 103)(17 76 43 102)(18 75 44 101)(19 74 45 100)(20 73 46 99)(21 72 47 98)(22 71 48 97)(23 70 49 96)(24 69 50 95)(25 68 51 94)(26 67 52 93)
(2 30 10)(3 7 19)(4 36 28)(5 13 37)(6 42 46)(8 48 12)(9 25 21)(11 31 39)(15 43 23)(16 20 32)(17 49 41)(18 26 50)(22 38 34)(24 44 52)(29 33 45)(35 51 47)(54 82 62)(55 59 71)(56 88 80)(57 65 89)(58 94 98)(60 100 64)(61 77 73)(63 83 91)(67 95 75)(68 72 84)(69 101 93)(70 78 102)(74 90 86)(76 96 104)(81 85 97)(87 103 99)

G:=sub<Sym(104)| (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)(53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104), (1,92,27,66)(2,91,28,65)(3,90,29,64)(4,89,30,63)(5,88,31,62)(6,87,32,61)(7,86,33,60)(8,85,34,59)(9,84,35,58)(10,83,36,57)(11,82,37,56)(12,81,38,55)(13,80,39,54)(14,79,40,53)(15,78,41,104)(16,77,42,103)(17,76,43,102)(18,75,44,101)(19,74,45,100)(20,73,46,99)(21,72,47,98)(22,71,48,97)(23,70,49,96)(24,69,50,95)(25,68,51,94)(26,67,52,93), (2,30,10)(3,7,19)(4,36,28)(5,13,37)(6,42,46)(8,48,12)(9,25,21)(11,31,39)(15,43,23)(16,20,32)(17,49,41)(18,26,50)(22,38,34)(24,44,52)(29,33,45)(35,51,47)(54,82,62)(55,59,71)(56,88,80)(57,65,89)(58,94,98)(60,100,64)(61,77,73)(63,83,91)(67,95,75)(68,72,84)(69,101,93)(70,78,102)(74,90,86)(76,96,104)(81,85,97)(87,103,99)>;

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)(53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104), (1,92,27,66)(2,91,28,65)(3,90,29,64)(4,89,30,63)(5,88,31,62)(6,87,32,61)(7,86,33,60)(8,85,34,59)(9,84,35,58)(10,83,36,57)(11,82,37,56)(12,81,38,55)(13,80,39,54)(14,79,40,53)(15,78,41,104)(16,77,42,103)(17,76,43,102)(18,75,44,101)(19,74,45,100)(20,73,46,99)(21,72,47,98)(22,71,48,97)(23,70,49,96)(24,69,50,95)(25,68,51,94)(26,67,52,93), (2,30,10)(3,7,19)(4,36,28)(5,13,37)(6,42,46)(8,48,12)(9,25,21)(11,31,39)(15,43,23)(16,20,32)(17,49,41)(18,26,50)(22,38,34)(24,44,52)(29,33,45)(35,51,47)(54,82,62)(55,59,71)(56,88,80)(57,65,89)(58,94,98)(60,100,64)(61,77,73)(63,83,91)(67,95,75)(68,72,84)(69,101,93)(70,78,102)(74,90,86)(76,96,104)(81,85,97)(87,103,99) );

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),(53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104)], [(1,92,27,66),(2,91,28,65),(3,90,29,64),(4,89,30,63),(5,88,31,62),(6,87,32,61),(7,86,33,60),(8,85,34,59),(9,84,35,58),(10,83,36,57),(11,82,37,56),(12,81,38,55),(13,80,39,54),(14,79,40,53),(15,78,41,104),(16,77,42,103),(17,76,43,102),(18,75,44,101),(19,74,45,100),(20,73,46,99),(21,72,47,98),(22,71,48,97),(23,70,49,96),(24,69,50,95),(25,68,51,94),(26,67,52,93)], [(2,30,10),(3,7,19),(4,36,28),(5,13,37),(6,42,46),(8,48,12),(9,25,21),(11,31,39),(15,43,23),(16,20,32),(17,49,41),(18,26,50),(22,38,34),(24,44,52),(29,33,45),(35,51,47),(54,82,62),(55,59,71),(56,88,80),(57,65,89),(58,94,98),(60,100,64),(61,77,73),(63,83,91),(67,95,75),(68,72,84),(69,101,93),(70,78,102),(74,90,86),(76,96,104),(81,85,97),(87,103,99)])

Matrix representation of Dic26⋊C3 in GL6(𝔽157)

13869115149450
1384613815300
134504130050
1384211942138153
9288123145873
1401711744136155
,
5398106109441
10010153155153110
1091524611740145
150611491054660
106155146644698
965545119354
,
896888215667
000010
100000
15628868892
000001
010000

G:=sub<GL(6,GF(157))| [138,138,134,138,92,140,69,46,50,42,88,17,115,138,4,119,123,117,149,153,130,42,145,44,4,0,0,138,8,136,50,0,50,153,73,155],[53,100,109,150,106,96,98,10,152,61,155,55,106,153,46,149,146,45,10,155,117,105,64,11,94,153,40,46,46,93,41,110,145,60,98,54],[89,0,1,156,0,0,68,0,0,2,0,1,88,0,0,88,0,0,2,0,0,68,0,0,156,1,0,89,0,0,67,0,0,2,1,0] >;

Dic26⋊C3 in GAP, Magma, Sage, TeX

{\rm Dic}_{26}\rtimes C_3
% in TeX

G:=Group("Dic26:C3");
// GroupNames label

G:=SmallGroup(312,8);
// by ID

G=gap.SmallGroup(312,8);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-13,60,141,66,7204,464]);
// Polycyclic

G:=Group<a,b,c|a^52=c^3=1,b^2=a^26,b*a*b^-1=a^-1,c*a*c^-1=a^9,b*c=c*b>;
// generators/relations

Export

Subgroup lattice of Dic26⋊C3 in TeX
Character table of Dic26⋊C3 in TeX

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