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G = Dic26⋊C4order 416 = 25·13

1st semidirect product of Dic26 and C4 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D26.2D4, Dic261C4, Dic13.21D4, C131C4≀C2, D42(C13⋊C4), (D4×C13)⋊2C4, C52.2(C2×C4), C52.C41C2, D42D13.2C2, C26.6(C22⋊C4), (C4×D13).8C22, C2.7(D13.D4), (C4×C13⋊C4)⋊1C2, C4.2(C2×C13⋊C4), SmallGroup(416,83)

Series: Derived Chief Lower central Upper central

C1C52 — Dic26⋊C4
C1C13C26Dic13C4×D13C52.C4 — Dic26⋊C4
C13C26C52 — Dic26⋊C4
C1C2C4D4

Generators and relations for Dic26⋊C4
 G = < a,b,c | a52=c4=1, b2=a26, bab-1=a-1, cac-1=a21, cbc-1=a39b >

4C2
26C2
2C22
13C4
13C22
26C4
26C4
26C4
2D13
4C26
13Q8
13C2×C4
26D4
26C2×C4
26C2×C4
26C8
2C2×C26
2C13⋊C4
2Dic13
2C13⋊C4
13C4○D4
13C42
13M4(2)
2C13⋊C8
2C2×Dic13
2C13⋊D4
2C2×C13⋊C4
13C4≀C2

Character table of Dic26⋊C4

 class 12A2B2C4A4B4C4D4E4F4G4H8A8B13A13B13C26A26B26C26D26E26F26G26H26I52A52B52C
 size 114262131326262626525252444444888888888
ρ111111111111111111111111111111    trivial
ρ211-111111111-1-1-1111111-1-1-1-1-1-1111    linear of order 2
ρ311-11111-1-1-1-1-111111111-1-1-1-1-1-1111    linear of order 2
ρ41111111-1-1-1-11-1-1111111111111111    linear of order 2
ρ5111-11-1-1-iii-i-1i-i111111111111111    linear of order 4
ρ611-1-11-1-1-iii-i1-ii111111-1-1-1-1-1-1111    linear of order 4
ρ7111-11-1-1i-i-ii-1-ii111111111111111    linear of order 4
ρ811-1-11-1-1i-i-ii1i-i111111-1-1-1-1-1-1111    linear of order 4
ρ92202-2-2-20000000222222000000-2-2-2    orthogonal lifted from D4
ρ10220-2-2220000000222222000000-2-2-2    orthogonal lifted from D4
ρ112-2000-2i2i1-i1+i-1-i-1+i000222-2-2-2000000000    complex lifted from C4≀C2
ρ122-20002i-2i1+i1-i-1+i-1-i000222-2-2-2000000000    complex lifted from C4≀C2
ρ132-2000-2i2i-1+i-1-i1+i1-i000222-2-2-2000000000    complex lifted from C4≀C2
ρ142-20002i-2i-1-i-1+i1-i1+i000222-2-2-2000000000    complex lifted from C4≀C2
ρ154400-4000000000ζ131213813513ζ139137136134ζ13111310133132ζ131213813513ζ13111310133132ζ139137136134ζ1311131013313213913713613413111310133132ζ139137136134131213813513ζ13121381351313111310133132131213813513139137136134    orthogonal lifted from D13.D4
ρ1644404000000000ζ131213813513ζ139137136134ζ13111310133132ζ131213813513ζ13111310133132ζ139137136134ζ13111310133132ζ139137136134ζ13111310133132ζ139137136134ζ131213813513ζ131213813513ζ13111310133132ζ131213813513ζ139137136134    orthogonal lifted from C13⋊C4
ρ174400-4000000000ζ13111310133132ζ131213813513ζ139137136134ζ13111310133132ζ139137136134ζ131213813513139137136134131213813513ζ139137136134ζ131213813513ζ131113101331321311131013313213913713613413111310133132131213813513    orthogonal lifted from D13.D4
ρ184400-4000000000ζ139137136134ζ13111310133132ζ131213813513ζ139137136134ζ131213813513ζ13111310133132131213813513ζ13111310133132ζ13121381351313111310133132139137136134ζ13913713613413121381351313913713613413111310133132    orthogonal lifted from D13.D4
ρ1944404000000000ζ13111310133132ζ131213813513ζ139137136134ζ13111310133132ζ139137136134ζ131213813513ζ139137136134ζ131213813513ζ139137136134ζ131213813513ζ13111310133132ζ13111310133132ζ139137136134ζ13111310133132ζ131213813513    orthogonal lifted from C13⋊C4
ρ2044-404000000000ζ139137136134ζ13111310133132ζ131213813513ζ139137136134ζ131213813513ζ131113101331321312138135131311131013313213121381351313111310133132139137136134139137136134ζ131213813513ζ139137136134ζ13111310133132    orthogonal lifted from C2×C13⋊C4
ρ214400-4000000000ζ13111310133132ζ131213813513ζ139137136134ζ13111310133132ζ139137136134ζ131213813513ζ139137136134ζ13121381351313913713613413121381351313111310133132ζ1311131013313213913713613413111310133132131213813513    orthogonal lifted from D13.D4
ρ2244404000000000ζ139137136134ζ13111310133132ζ131213813513ζ139137136134ζ131213813513ζ13111310133132ζ131213813513ζ13111310133132ζ131213813513ζ13111310133132ζ139137136134ζ139137136134ζ131213813513ζ139137136134ζ13111310133132    orthogonal lifted from C13⋊C4
ρ2344-404000000000ζ13111310133132ζ131213813513ζ139137136134ζ13111310133132ζ139137136134ζ1312138135131391371361341312138135131391371361341312138135131311131013313213111310133132ζ139137136134ζ13111310133132ζ131213813513    orthogonal lifted from C2×C13⋊C4
ρ244400-4000000000ζ131213813513ζ139137136134ζ13111310133132ζ131213813513ζ13111310133132ζ13913713613413111310133132ζ139137136134ζ13111310133132139137136134ζ13121381351313121381351313111310133132131213813513139137136134    orthogonal lifted from D13.D4
ρ2544-404000000000ζ131213813513ζ139137136134ζ13111310133132ζ131213813513ζ13111310133132ζ1391371361341311131013313213913713613413111310133132139137136134131213813513131213813513ζ13111310133132ζ131213813513ζ139137136134    orthogonal lifted from C2×C13⋊C4
ρ264400-4000000000ζ139137136134ζ13111310133132ζ131213813513ζ139137136134ζ131213813513ζ13111310133132ζ13121381351313111310133132131213813513ζ13111310133132ζ13913713613413913713613413121381351313913713613413111310133132    orthogonal lifted from D13.D4
ρ278-8000000000000139+2ζ137+2ζ136+2ζ1341311+2ζ1310+2ζ133+2ζ1321312+2ζ138+2ζ135+2ζ13-2ζ139-2ζ137-2ζ136-2ζ134-2ζ1312-2ζ138-2ζ135-2ζ13-2ζ1311-2ζ1310-2ζ133-2ζ132000000000    symplectic faithful, Schur index 2
ρ288-80000000000001311+2ζ1310+2ζ133+2ζ1321312+2ζ138+2ζ135+2ζ13139+2ζ137+2ζ136+2ζ134-2ζ1311-2ζ1310-2ζ133-2ζ132-2ζ139-2ζ137-2ζ136-2ζ134-2ζ1312-2ζ138-2ζ135-2ζ13000000000    symplectic faithful, Schur index 2
ρ298-80000000000001312+2ζ138+2ζ135+2ζ13139+2ζ137+2ζ136+2ζ1341311+2ζ1310+2ζ133+2ζ132-2ζ1312-2ζ138-2ζ135-2ζ13-2ζ1311-2ζ1310-2ζ133-2ζ132-2ζ139-2ζ137-2ζ136-2ζ134000000000    symplectic faithful, Schur index 2

Smallest permutation representation of Dic26⋊C4
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 53 27 79)(2 104 28 78)(3 103 29 77)(4 102 30 76)(5 101 31 75)(6 100 32 74)(7 99 33 73)(8 98 34 72)(9 97 35 71)(10 96 36 70)(11 95 37 69)(12 94 38 68)(13 93 39 67)(14 92 40 66)(15 91 41 65)(16 90 42 64)(17 89 43 63)(18 88 44 62)(19 87 45 61)(20 86 46 60)(21 85 47 59)(22 84 48 58)(23 83 49 57)(24 82 50 56)(25 81 51 55)(26 80 52 54)
(1 27)(2 32 26 48)(3 37 51 17)(4 42 24 38)(5 47 49 7)(6 52 22 28)(8 10 20 18)(9 15 45 39)(11 25 43 29)(12 30 16 50)(13 35 41 19)(14 40)(21 23 33 31)(34 36 46 44)(53 66 79 92)(54 71 104 61)(55 76 77 82)(56 81 102 103)(57 86 75 72)(58 91 100 93)(59 96 73 62)(60 101 98 83)(63 64 69 94)(65 74 67 84)(68 89 90 95)(70 99 88 85)(78 87 80 97)

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

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

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

Matrix representation of Dic26⋊C4 in GL6(𝔽313)

2500000
02880000
0029100
00311010
00241001
0013910324072
,
02880000
28800000
0028499212101
0075297286100
002176214729
003613436211
,
31200000
02880000
002270285283
00181701413
0016910413843
0020843269103

G:=sub<GL(6,GF(313))| [25,0,0,0,0,0,0,288,0,0,0,0,0,0,29,311,241,139,0,0,1,0,0,103,0,0,0,1,0,240,0,0,0,0,1,72],[0,288,0,0,0,0,288,0,0,0,0,0,0,0,284,75,217,36,0,0,99,297,62,134,0,0,212,286,147,36,0,0,101,100,29,211],[312,0,0,0,0,0,0,288,0,0,0,0,0,0,2,181,169,208,0,0,270,70,104,43,0,0,285,141,138,269,0,0,283,3,43,103] >;

Dic26⋊C4 in GAP, Magma, Sage, TeX

{\rm Dic}_{26}\rtimes C_4
% in TeX

G:=Group("Dic26:C4");
// GroupNames label

G:=SmallGroup(416,83);
// by ID

G=gap.SmallGroup(416,83);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,86,579,297,69,9221,3473]);
// Polycyclic

G:=Group<a,b,c|a^52=c^4=1,b^2=a^26,b*a*b^-1=a^-1,c*a*c^-1=a^21,c*b*c^-1=a^39*b>;
// generators/relations

Export

Subgroup lattice of Dic26⋊C4 in TeX
Character table of Dic26⋊C4 in TeX

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