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G = D42D13order 208 = 24·13

The semidirect product of D4 and D13 acting through Inn(D4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D42D13, C4.5D26, Dic263C2, C52.5C22, C26.6C23, C22.1D26, D26.2C22, Dic13.8C22, (C4×D13)⋊2C2, (D4×C13)⋊3C2, C132(C4○D4), C13⋊D42C2, (C2×C26).C22, (C2×Dic13)⋊3C2, C2.7(C22×D13), SmallGroup(208,40)

Series: Derived Chief Lower central Upper central

C1C26 — D42D13
C1C13C26D26C4×D13 — D42D13
C13C26 — D42D13
C1C2D4

Generators and relations for D42D13
 G = < a,b,c,d | a4=b2=c13=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >

2C2
2C2
26C2
13C4
13C22
13C4
13C4
2C26
2C26
2D13
13C2×C4
13D4
13D4
13C2×C4
13Q8
13C2×C4
13C4○D4

Smallest permutation representation of D42D13
On 104 points
Generators in S104
(1 77 16 53)(2 78 17 54)(3 66 18 55)(4 67 19 56)(5 68 20 57)(6 69 21 58)(7 70 22 59)(8 71 23 60)(9 72 24 61)(10 73 25 62)(11 74 26 63)(12 75 14 64)(13 76 15 65)(27 98 48 86)(28 99 49 87)(29 100 50 88)(30 101 51 89)(31 102 52 90)(32 103 40 91)(33 104 41 79)(34 92 42 80)(35 93 43 81)(36 94 44 82)(37 95 45 83)(38 96 46 84)(39 97 47 85)
(1 81)(2 82)(3 83)(4 84)(5 85)(6 86)(7 87)(8 88)(9 89)(10 90)(11 91)(12 79)(13 80)(14 104)(15 92)(16 93)(17 94)(18 95)(19 96)(20 97)(21 98)(22 99)(23 100)(24 101)(25 102)(26 103)(27 58)(28 59)(29 60)(30 61)(31 62)(32 63)(33 64)(34 65)(35 53)(36 54)(37 55)(38 56)(39 57)(40 74)(41 75)(42 76)(43 77)(44 78)(45 66)(46 67)(47 68)(48 69)(49 70)(50 71)(51 72)(52 73)
(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 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 17)(15 16)(18 26)(19 25)(20 24)(21 23)(27 50)(28 49)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)(37 40)(38 52)(39 51)(53 65)(54 64)(55 63)(56 62)(57 61)(58 60)(66 74)(67 73)(68 72)(69 71)(75 78)(76 77)(79 94)(80 93)(81 92)(82 104)(83 103)(84 102)(85 101)(86 100)(87 99)(88 98)(89 97)(90 96)(91 95)

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

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

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

D42D13 is a maximal subgroup of
Dic26⋊C4  D8⋊D13  D83D13  D4.D26  D26.6D4  Dic26.C4  D46D26  C4○D4×D13  D4.10D26
D42D13 is a maximal quotient of
C23.11D26  C22⋊Dic26  C23.D26  Dic134D4  D26.12D4  C23.6D26  C22.D52  Dic133Q8  Dic13.Q8  C4.Dic26  C4⋊C47D13  D262Q8  C4⋊C4⋊D13  D4×Dic13  C23.18D26  C52.17D4  C522D4  Dic13⋊D4

40 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E13A···13F26A···26F26G···26R52A···52F
order122224444413···1326···2626···2652···52
size1122262131326262···22···24···44···4

40 irreducible representations

dim11111122224
type+++++++++-
imageC1C2C2C2C2C2C4○D4D13D26D26D42D13
kernelD42D13Dic26C4×D13C2×Dic13C13⋊D4D4×C13C13D4C4C22C1
# reps111221266126

Matrix representation of D42D13 in GL4(𝔽53) generated by

30000
02300
00520
00052
,
02300
30000
0010
0001
,
1000
0100
00441
003120
,
1000
05200
004611
00397
G:=sub<GL(4,GF(53))| [30,0,0,0,0,23,0,0,0,0,52,0,0,0,0,52],[0,30,0,0,23,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,44,31,0,0,1,20],[1,0,0,0,0,52,0,0,0,0,46,39,0,0,11,7] >;

D42D13 in GAP, Magma, Sage, TeX

D_4\rtimes_2D_{13}
% in TeX

G:=Group("D4:2D13");
// GroupNames label

G:=SmallGroup(208,40);
// by ID

G=gap.SmallGroup(208,40);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-13,46,182,97,4804]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^13=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations

Export

Subgroup lattice of D42D13 in TeX

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