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G = D52⋊C4order 416 = 25·13

2nd semidirect product of D52 and C4 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D522C4, D26.3D4, Dic13.22D4, C132C4≀C2, C52.4(C2×C4), Q82(C13⋊C4), (Q8×C13)⋊2C4, C52.C42C2, D52⋊C2.2C2, C26.8(C22⋊C4), C2.9(D13.D4), (C4×D13).10C22, (C4×C13⋊C4)⋊2C2, C4.4(C2×C13⋊C4), SmallGroup(416,85)

Series: Derived Chief Lower central Upper central

C1C52 — D52⋊C4
C1C13C26Dic13C4×D13C52.C4 — D52⋊C4
C13C26C52 — D52⋊C4
C1C2C4Q8

Generators and relations for D52⋊C4
 G = < a,b,c | a52=b2=c4=1, bab=a-1, cac-1=a21, cbc-1=a7b >

26C2
52C2
2C4
13C22
13C4
26C4
26C22
26C4
2D13
4D13
13D4
13C2×C4
26D4
26C2×C4
26C2×C4
26C8
2C52
2C13⋊C4
2D26
2C13⋊C4
13C4○D4
13C42
13M4(2)
2C13⋊C8
2C4×D13
2D52
2C2×C13⋊C4
13C4≀C2

Character table of D52⋊C4

 class 12A2B2C4A4B4C4D4E4F4G4H8A8B13A13B13C26A26B26C52A52B52C52D52E52F52G52H52I
 size 112652241313262626265252444444888888888
ρ111111111111111111111111111111    trivial
ρ2111-11-1111111-1-1111111-1-111-1-1-11-1    linear of order 2
ρ311111111-1-1-1-1-1-1111111111111111    linear of order 2
ρ4111-11-111-1-1-1-111111111-1-111-1-1-11-1    linear of order 2
ρ511-1-111-1-1i-i-iii-i111111111111111    linear of order 4
ρ611-111-1-1-1i-i-ii-ii111111-1-111-1-1-11-1    linear of order 4
ρ711-1-111-1-1-iii-i-ii111111111111111    linear of order 4
ρ811-111-1-1-1-iii-ii-i111111-1-111-1-1-11-1    linear of order 4
ρ922-20-202200000022222200-2-2000-20    orthogonal lifted from D4
ρ102220-20-2-200000022222200-2-2000-20    orthogonal lifted from D4
ρ112-20000-2i2i-1-i1-i-1+i1+i00222-2-2-2000000000    complex lifted from C4≀C2
ρ122-20000-2i2i1+i-1+i1-i-1-i00222-2-2-2000000000    complex lifted from C4≀C2
ρ132-200002i-2i1-i-1-i1+i-1+i00222-2-2-2000000000    complex lifted from C4≀C2
ρ142-200002i-2i-1+i1+i-1-i1-i00222-2-2-2000000000    complex lifted from C4≀C2
ρ1544004-400000000ζ13111310133132ζ139137136134ζ131213813513ζ139137136134ζ131213813513ζ1311131013313213121381351313111310133132ζ131213813513ζ1311131013313213121381351313111310133132139137136134ζ139137136134139137136134    orthogonal lifted from C2×C13⋊C4
ρ164400-4000000000ζ13111310133132ζ139137136134ζ131213813513ζ139137136134ζ131213813513ζ13111310133132ζ1312138135131311131013313213121381351313111310133132131213813513ζ13111310133132ζ139137136134139137136134139137136134    orthogonal lifted from D13.D4
ρ174400-4000000000ζ131213813513ζ13111310133132ζ139137136134ζ13111310133132ζ139137136134ζ131213813513139137136134131213813513139137136134131213813513ζ139137136134ζ131213813513ζ131113101331321311131013313213111310133132    orthogonal lifted from D13.D4
ρ184400-4000000000ζ139137136134ζ131213813513ζ13111310133132ζ131213813513ζ13111310133132ζ139137136134ζ131113101331321391371361341311131013313213913713613413111310133132ζ139137136134131213813513131213813513ζ131213813513    orthogonal lifted from D13.D4
ρ1944004400000000ζ131213813513ζ13111310133132ζ139137136134ζ13111310133132ζ139137136134ζ131213813513ζ139137136134ζ131213813513ζ139137136134ζ131213813513ζ139137136134ζ131213813513ζ13111310133132ζ13111310133132ζ13111310133132    orthogonal lifted from C13⋊C4
ρ2044004400000000ζ13111310133132ζ139137136134ζ131213813513ζ139137136134ζ131213813513ζ13111310133132ζ131213813513ζ13111310133132ζ131213813513ζ13111310133132ζ131213813513ζ13111310133132ζ139137136134ζ139137136134ζ139137136134    orthogonal lifted from C13⋊C4
ρ2144004-400000000ζ139137136134ζ131213813513ζ13111310133132ζ131213813513ζ13111310133132ζ13913713613413111310133132139137136134ζ13111310133132ζ13913713613413111310133132139137136134131213813513ζ131213813513131213813513    orthogonal lifted from C2×C13⋊C4
ρ224400-4000000000ζ139137136134ζ131213813513ζ13111310133132ζ131213813513ζ13111310133132ζ13913713613413111310133132ζ13913713613413111310133132139137136134ζ13111310133132139137136134ζ131213813513131213813513131213813513    orthogonal lifted from D13.D4
ρ234400-4000000000ζ13111310133132ζ139137136134ζ131213813513ζ139137136134ζ131213813513ζ13111310133132131213813513ζ1311131013313213121381351313111310133132ζ13121381351313111310133132139137136134139137136134ζ139137136134    orthogonal lifted from D13.D4
ρ244400-4000000000ζ131213813513ζ13111310133132ζ139137136134ζ13111310133132ζ139137136134ζ131213813513ζ139137136134ζ1312138135131391371361341312138135131391371361341312138135131311131013313213111310133132ζ13111310133132    orthogonal lifted from D13.D4
ρ2544004-400000000ζ131213813513ζ13111310133132ζ139137136134ζ13111310133132ζ139137136134ζ131213813513139137136134131213813513ζ139137136134ζ13121381351313913713613413121381351313111310133132ζ1311131013313213111310133132    orthogonal lifted from C2×C13⋊C4
ρ2644004400000000ζ139137136134ζ131213813513ζ13111310133132ζ131213813513ζ13111310133132ζ139137136134ζ13111310133132ζ139137136134ζ13111310133132ζ139137136134ζ13111310133132ζ139137136134ζ131213813513ζ131213813513ζ131213813513    orthogonal lifted from C13⋊C4
ρ278-80000000000001312+2ζ138+2ζ135+2ζ131311+2ζ1310+2ζ133+2ζ132139+2ζ137+2ζ136+2ζ134-2ζ1311-2ζ1310-2ζ133-2ζ132-2ζ139-2ζ137-2ζ136-2ζ134-2ζ1312-2ζ138-2ζ135-2ζ13000000000    orthogonal faithful
ρ288-8000000000000139+2ζ137+2ζ136+2ζ1341312+2ζ138+2ζ135+2ζ131311+2ζ1310+2ζ133+2ζ132-2ζ1312-2ζ138-2ζ135-2ζ13-2ζ1311-2ζ1310-2ζ133-2ζ132-2ζ139-2ζ137-2ζ136-2ζ134000000000    orthogonal faithful
ρ298-80000000000001311+2ζ1310+2ζ133+2ζ132139+2ζ137+2ζ136+2ζ1341312+2ζ138+2ζ135+2ζ13-2ζ139-2ζ137-2ζ136-2ζ134-2ζ1312-2ζ138-2ζ135-2ζ13-2ζ1311-2ζ1310-2ζ133-2ζ132000000000    orthogonal faithful

Smallest permutation representation of D52⋊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 98)(2 97)(3 96)(4 95)(5 94)(6 93)(7 92)(8 91)(9 90)(10 89)(11 88)(12 87)(13 86)(14 85)(15 84)(16 83)(17 82)(18 81)(19 80)(20 79)(21 78)(22 77)(23 76)(24 75)(25 74)(26 73)(27 72)(28 71)(29 70)(30 69)(31 68)(32 67)(33 66)(34 65)(35 64)(36 63)(37 62)(38 61)(39 60)(40 59)(41 58)(42 57)(43 56)(44 55)(45 54)(46 53)(47 104)(48 103)(49 102)(50 101)(51 100)(52 99)
(1 40 27 14)(2 45 52 35)(3 50 25 4)(5 8 23 46)(6 13 48 15)(7 18 21 36)(9 28 19 26)(10 33 44 47)(11 38 17 16)(12 43 42 37)(20 31 34 49)(22 41 32 39)(24 51 30 29)(53 77 93 69)(54 82 66 90)(55 87 91 59)(56 92 64 80)(57 97 89 101)(58 102 62 70)(61 65 85 81)(63 75 83 71)(67 95 79 103)(68 100 104 72)(74 78 98 94)(76 88 96 84)

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

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

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

Matrix representation of D52⋊C4 in GL6(𝔽313)

2500000
22880000
00000312
0013024330
0028340252282
003127461243
,
1581190000
2401550000
0024131104172
0007214172
0072141720
0017210431241
,
28800000
2410000
00312000
0030313130
00000312
0028225240283

G:=sub<GL(6,GF(313))| [25,2,0,0,0,0,0,288,0,0,0,0,0,0,0,1,283,31,0,0,0,30,40,274,0,0,0,243,252,61,0,0,312,30,282,243],[158,240,0,0,0,0,119,155,0,0,0,0,0,0,241,0,72,172,0,0,31,72,141,104,0,0,104,141,72,31,0,0,172,72,0,241],[288,24,0,0,0,0,0,1,0,0,0,0,0,0,312,30,0,282,0,0,0,31,0,252,0,0,0,31,0,40,0,0,0,30,312,283] >;

D52⋊C4 in GAP, Magma, Sage, TeX

D_{52}\rtimes C_4
% in TeX

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

G:=SmallGroup(416,85);
// by ID

G=gap.SmallGroup(416,85);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,24,121,103,86,579,297,69,9221,3473]);
// Polycyclic

G:=Group<a,b,c|a^52=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^21,c*b*c^-1=a^7*b>;
// generators/relations

Export

Subgroup lattice of D52⋊C4 in TeX
Character table of D52⋊C4 in TeX

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