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G = D46D26order 416 = 25·13

2nd semidirect product of D4 and D26 acting through Inn(D4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D46D26, C232D26, D528C22, C26.7C24, C52.21C23, D26.3C23, C1312+ 1+4, Dic268C22, Dic13.4C23, (C2×C4)⋊3D26, (D4×C26)⋊7C2, (C2×D4)⋊7D13, (D4×D13)⋊4C2, (C2×C52)⋊3C22, D42D134C2, D525C25C2, (D4×C13)⋊7C22, (C4×D13)⋊1C22, C13⋊D43C22, (C2×C26).2C23, C2.8(C23×D13), (C22×C26)⋊5C22, C4.21(C22×D13), (C2×Dic13)⋊4C22, (C22×D13)⋊3C22, C22.6(C22×D13), (C2×C13⋊D4)⋊11C2, SmallGroup(416,218)

Series: Derived Chief Lower central Upper central

C1C26 — D46D26
C1C13C26D26C22×D13D4×D13 — D46D26
C13C26 — D46D26
C1C2C2×D4

Generators and relations for D46D26
 G = < a,b,c,d | a4=b2=c26=d2=1, bab=cac-1=a-1, ad=da, cbc-1=dbd=a2b, dcd=c-1 >

Subgroups: 1120 in 166 conjugacy classes, 85 normal (11 characteristic)
C1, C2, C2 [×9], C4 [×2], C4 [×4], C22, C22 [×4], C22 [×10], C2×C4, C2×C4 [×8], D4 [×4], D4 [×14], Q8 [×2], C23 [×2], C23 [×4], C13, C2×D4, C2×D4 [×8], C4○D4 [×6], D13 [×4], C26, C26 [×5], 2+ 1+4, Dic13 [×4], C52 [×2], D26 [×4], D26 [×4], C2×C26, C2×C26 [×4], C2×C26 [×2], Dic26 [×2], C4×D13 [×4], D52 [×2], C2×Dic13 [×4], C13⋊D4 [×12], C2×C52, D4×C13 [×4], C22×D13 [×4], C22×C26 [×2], D525C2 [×2], D4×D13 [×4], D42D13 [×4], C2×C13⋊D4 [×4], D4×C26, D46D26
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C24, D13, 2+ 1+4, D26 [×7], C22×D13 [×7], C23×D13, D46D26

Smallest permutation representation of D46D26
On 104 points
Generators in S104
(1 89 15 102)(2 103 16 90)(3 91 17 104)(4 79 18 92)(5 93 19 80)(6 81 20 94)(7 95 21 82)(8 83 22 96)(9 97 23 84)(10 85 24 98)(11 99 25 86)(12 87 26 100)(13 101 14 88)(27 73 47 60)(28 61 48 74)(29 75 49 62)(30 63 50 76)(31 77 51 64)(32 65 52 78)(33 53 40 66)(34 67 41 54)(35 55 42 68)(36 69 43 56)(37 57 44 70)(38 71 45 58)(39 59 46 72)
(1 78)(2 66)(3 54)(4 68)(5 56)(6 70)(7 58)(8 72)(9 60)(10 74)(11 62)(12 76)(13 64)(14 77)(15 65)(16 53)(17 67)(18 55)(19 69)(20 57)(21 71)(22 59)(23 73)(24 61)(25 75)(26 63)(27 84)(28 98)(29 86)(30 100)(31 88)(32 102)(33 90)(34 104)(35 92)(36 80)(37 94)(38 82)(39 96)(40 103)(41 91)(42 79)(43 93)(44 81)(45 95)(46 83)(47 97)(48 85)(49 99)(50 87)(51 101)(52 89)
(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 8)(2 7)(3 6)(4 5)(9 13)(10 12)(14 23)(15 22)(16 21)(17 20)(18 19)(24 26)(27 51)(28 50)(29 49)(30 48)(31 47)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 40)(39 52)(53 58)(54 57)(55 56)(59 78)(60 77)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)(79 93)(80 92)(81 91)(82 90)(83 89)(84 88)(85 87)(94 104)(95 103)(96 102)(97 101)(98 100)

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

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

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

77 conjugacy classes

class 1 2A2B···2F2G2H2I2J4A4B4C4D4E4F13A···13F26A···26R26S···26AP52A···52L
order122···2222244444413···1326···2626···2652···52
size112···22626262622262626262···22···24···44···4

77 irreducible representations

dim111111222244
type+++++++++++
imageC1C2C2C2C2C2D13D26D26D262+ 1+4D46D26
kernelD46D26D525C2D4×D13D42D13C2×C13⋊D4D4×C26C2×D4C2×C4D4C23C13C1
# reps124441662412112

Matrix representation of D46D26 in GL4(𝔽53) generated by

001335
002740
401800
261300
,
0010
0001
1000
0100
,
9600
44000
004447
0090
,
411400
391200
001239
001441
G:=sub<GL(4,GF(53))| [0,0,40,26,0,0,18,13,13,27,0,0,35,40,0,0],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[9,44,0,0,6,0,0,0,0,0,44,9,0,0,47,0],[41,39,0,0,14,12,0,0,0,0,12,14,0,0,39,41] >;

D46D26 in GAP, Magma, Sage, TeX

D_4\rtimes_6D_{26}
% in TeX

G:=Group("D4:6D26");
// GroupNames label

G:=SmallGroup(416,218);
// by ID

G=gap.SmallGroup(416,218);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,188,579,13829]);
// Polycyclic

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

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