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G = D521C4order 416 = 25·13

1st semidirect product of D52 and C4 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D521C4, D13.2D8, D26.18D4, Dic13.2D4, D13.2SD16, D13⋊C81C2, C13⋊(D4⋊C4), D41(C13⋊C4), (D4×C13)⋊1C4, C52.1(C2×C4), C52⋊C41C2, (D4×D13).2C2, C26.5(C22⋊C4), (C4×D13).7C22, C2.6(D13.D4), C4.1(C2×C13⋊C4), SmallGroup(416,82)

Series: Derived Chief Lower central Upper central

C1C52 — D521C4
C1C13C26D26C4×D13C52⋊C4 — D521C4
C13C26C52 — D521C4
C1C2C4D4

Generators and relations for D521C4
 G = < a,b,c | a52=b2=c4=1, bab=a-1, cac-1=a31, cbc-1=a17b >

4C2
13C2
13C2
52C2
2C22
13C4
13C22
26C22
52C22
52C22
52C4
4D13
4C26
13D4
13C2×C4
26D4
26C23
26C2×C4
26C8
2C2×C26
2D26
4D26
4C13⋊C4
4D26
13C4⋊C4
13C2×D4
13C2×C8
2C13⋊C8
2C13⋊D4
2C2×C13⋊C4
2C22×D13
13D4⋊C4

Character table of D521C4

 class 12A2B2C2D2E4A4B4C4D8A8B8C8D13A13B13C26A26B26C26D26E26F26G26H26I52A52B52C
 size 114131352226525226262626444444888888888
ρ111111111111111111111111111111    trivial
ρ211-111-11111-1-1-1-1111111-1-1-1-1-1-1111    linear of order 2
ρ311111111-1-1-1-1-1-1111111111111111    linear of order 2
ρ411-111-111-1-11111111111-1-1-1-1-1-1111    linear of order 2
ρ5111-1-1-11-1-ii-iii-i111111111111111    linear of order 4
ρ611-1-1-111-1-iii-i-ii111111-1-1-1-1-1-1111    linear of order 4
ρ7111-1-1-11-1i-ii-i-ii111111111111111    linear of order 4
ρ811-1-1-111-1i-i-iii-i111111-1-1-1-1-1-1111    linear of order 4
ρ9220-2-20-22000000222222000000-2-2-2    orthogonal lifted from D4
ρ10220220-2-2000000222222000000-2-2-2    orthogonal lifted from D4
ρ112-202-2000002-22-2222-2-2-2000000000    orthogonal lifted from D8
ρ122-202-200000-22-22222-2-2-2000000000    orthogonal lifted from D8
ρ132-20-2200000-2-2--2--2222-2-2-2000000000    complex lifted from SD16
ρ142-20-2200000--2--2-2-2222-2-2-2000000000    complex lifted from SD16
ρ1544-400040000000ζ131213813513ζ13111310133132ζ139137136134ζ131213813513ζ13111310133132ζ1391371361341312138135131311131013313213121381351313111310133132139137136134139137136134ζ13111310133132ζ139137136134ζ131213813513    orthogonal lifted from C2×C13⋊C4
ρ16440000-40000000ζ139137136134ζ131213813513ζ13111310133132ζ139137136134ζ131213813513ζ13111310133132139137136134131213813513ζ139137136134ζ131213813513ζ131113101331321311131013313213121381351313111310133132139137136134    orthogonal lifted from D13.D4
ρ1744400040000000ζ131213813513ζ13111310133132ζ139137136134ζ131213813513ζ13111310133132ζ139137136134ζ131213813513ζ13111310133132ζ131213813513ζ13111310133132ζ139137136134ζ139137136134ζ13111310133132ζ139137136134ζ131213813513    orthogonal lifted from C13⋊C4
ρ1844-400040000000ζ13111310133132ζ139137136134ζ131213813513ζ13111310133132ζ139137136134ζ1312138135131311131013313213913713613413111310133132139137136134131213813513131213813513ζ139137136134ζ131213813513ζ13111310133132    orthogonal lifted from C2×C13⋊C4
ρ19440000-40000000ζ139137136134ζ131213813513ζ13111310133132ζ139137136134ζ131213813513ζ13111310133132ζ139137136134ζ13121381351313913713613413121381351313111310133132ζ1311131013313213121381351313111310133132139137136134    orthogonal lifted from D13.D4
ρ20440000-40000000ζ13111310133132ζ139137136134ζ131213813513ζ13111310133132ζ139137136134ζ131213813513ζ1311131013313213913713613413111310133132ζ139137136134131213813513ζ13121381351313913713613413121381351313111310133132    orthogonal lifted from D13.D4
ρ21440000-40000000ζ131213813513ζ13111310133132ζ139137136134ζ131213813513ζ13111310133132ζ139137136134ζ13121381351313111310133132131213813513ζ13111310133132ζ13913713613413913713613413111310133132139137136134131213813513    orthogonal lifted from D13.D4
ρ2244400040000000ζ13111310133132ζ139137136134ζ131213813513ζ13111310133132ζ139137136134ζ131213813513ζ13111310133132ζ139137136134ζ13111310133132ζ139137136134ζ131213813513ζ131213813513ζ139137136134ζ131213813513ζ13111310133132    orthogonal lifted from C13⋊C4
ρ2344-400040000000ζ139137136134ζ131213813513ζ13111310133132ζ139137136134ζ131213813513ζ131113101331321391371361341312138135131391371361341312138135131311131013313213111310133132ζ131213813513ζ13111310133132ζ139137136134    orthogonal lifted from C2×C13⋊C4
ρ2444400040000000ζ139137136134ζ131213813513ζ13111310133132ζ139137136134ζ131213813513ζ13111310133132ζ139137136134ζ131213813513ζ139137136134ζ131213813513ζ13111310133132ζ13111310133132ζ131213813513ζ13111310133132ζ139137136134    orthogonal lifted from C13⋊C4
ρ25440000-40000000ζ131213813513ζ13111310133132ζ139137136134ζ131213813513ζ13111310133132ζ139137136134131213813513ζ13111310133132ζ13121381351313111310133132139137136134ζ13913713613413111310133132139137136134131213813513    orthogonal lifted from D13.D4
ρ26440000-40000000ζ13111310133132ζ139137136134ζ131213813513ζ13111310133132ζ139137136134ζ13121381351313111310133132ζ139137136134ζ13111310133132139137136134ζ13121381351313121381351313913713613413121381351313111310133132    orthogonal lifted from D13.D4
ρ278-80000000000001311+2ζ1310+2ζ133+2ζ132139+2ζ137+2ζ136+2ζ1341312+2ζ138+2ζ135+2ζ13-2ζ1311-2ζ1310-2ζ133-2ζ132-2ζ139-2ζ137-2ζ136-2ζ134-2ζ1312-2ζ138-2ζ135-2ζ13000000000    orthogonal faithful, Schur index 2
ρ288-8000000000000139+2ζ137+2ζ136+2ζ1341312+2ζ138+2ζ135+2ζ131311+2ζ1310+2ζ133+2ζ132-2ζ139-2ζ137-2ζ136-2ζ134-2ζ1312-2ζ138-2ζ135-2ζ13-2ζ1311-2ζ1310-2ζ133-2ζ132000000000    orthogonal faithful, Schur index 2
ρ298-80000000000001312+2ζ138+2ζ135+2ζ131311+2ζ1310+2ζ133+2ζ132139+2ζ137+2ζ136+2ζ134-2ζ1312-2ζ138-2ζ135-2ζ13-2ζ1311-2ζ1310-2ζ133-2ζ132-2ζ139-2ζ137-2ζ136-2ζ134000000000    orthogonal faithful, Schur index 2

Smallest permutation representation of D521C4
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 74)(2 73)(3 72)(4 71)(5 70)(6 69)(7 68)(8 67)(9 66)(10 65)(11 64)(12 63)(13 62)(14 61)(15 60)(16 59)(17 58)(18 57)(19 56)(20 55)(21 54)(22 53)(23 104)(24 103)(25 102)(26 101)(27 100)(28 99)(29 98)(30 97)(31 96)(32 95)(33 94)(34 93)(35 92)(36 91)(37 90)(38 89)(39 88)(40 87)(41 86)(42 85)(43 84)(44 83)(45 82)(46 81)(47 80)(48 79)(49 78)(50 77)(51 76)(52 75)
(2 48 26 32)(3 43 51 11)(4 38 24 42)(5 33 49 21)(6 28 22 52)(7 23 47 31)(8 18 20 10)(9 13 45 41)(12 50 16 30)(14 40)(15 35 39 19)(17 25 37 29)(34 44 46 36)(53 94 97 82)(54 89 70 61)(55 84 95 92)(56 79 68 71)(57 74 93 102)(58 69 66 81)(59 64 91 60)(62 101)(63 96 87 80)(65 86 85 90)(67 76 83 100)(72 103 104 99)(73 98 77 78)(75 88)

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

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

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

Matrix representation of D521C4 in GL6(𝔽313)

12250000
2493120000
00001312
002422823271
0032252273311
002422832101
,
02720000
22900000
00137100213176
00237076176
00272411000
0014823031276
,
28800000
35250000
007131282242
0031230712
000100
007130312212

G:=sub<GL(6,GF(313))| [1,249,0,0,0,0,225,312,0,0,0,0,0,0,0,242,32,242,0,0,0,282,252,283,0,0,1,32,273,2,0,0,312,71,311,101],[0,229,0,0,0,0,272,0,0,0,0,0,0,0,137,237,272,148,0,0,100,0,41,230,0,0,213,76,100,312,0,0,176,176,0,76],[288,35,0,0,0,0,0,25,0,0,0,0,0,0,71,312,0,71,0,0,31,30,1,30,0,0,282,71,0,312,0,0,242,2,0,212] >;

D521C4 in GAP, Magma, Sage, TeX

D_{52}\rtimes_1C_4
% in TeX

G:=Group("D52:1C4");
// GroupNames label

G:=SmallGroup(416,82);
// by ID

G=gap.SmallGroup(416,82);
# by ID

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

G:=Group<a,b,c|a^52=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^31,c*b*c^-1=a^17*b>;
// generators/relations

Export

Subgroup lattice of D521C4 in TeX
Character table of D521C4 in TeX

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