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G = D4.D13order 208 = 24·13

The non-split extension by D4 of D13 acting via D13/C13=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.D13, C4.2D26, C26.8D4, C132SD16, Dic262C2, C52.2C22, C132C82C2, (D4×C13).1C2, C2.5(C13⋊D4), SmallGroup(208,16)

Series: Derived Chief Lower central Upper central

C1C52 — D4.D13
C1C13C26C52Dic26 — D4.D13
C13C26C52 — D4.D13
C1C2C4D4

Generators and relations for D4.D13
 G = < a,b,c,d | a4=b2=c13=1, d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c-1 >

4C2
2C22
26C4
4C26
13C8
13Q8
2Dic13
2C2×C26
13SD16

Smallest permutation representation of D4.D13
On 104 points
Generators in S104
(1 51 21 37)(2 52 22 38)(3 40 23 39)(4 41 24 27)(5 42 25 28)(6 43 26 29)(7 44 14 30)(8 45 15 31)(9 46 16 32)(10 47 17 33)(11 48 18 34)(12 49 19 35)(13 50 20 36)(53 79 66 92)(54 80 67 93)(55 81 68 94)(56 82 69 95)(57 83 70 96)(58 84 71 97)(59 85 72 98)(60 86 73 99)(61 87 74 100)(62 88 75 101)(63 89 76 102)(64 90 77 103)(65 91 78 104)
(1 37)(2 38)(3 39)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 33)(11 34)(12 35)(13 36)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 51)(22 52)(23 40)(24 41)(25 42)(26 43)(53 66)(54 67)(55 68)(56 69)(57 70)(58 71)(59 72)(60 73)(61 74)(62 75)(63 76)(64 77)(65 78)
(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 78 21 65)(2 77 22 64)(3 76 23 63)(4 75 24 62)(5 74 25 61)(6 73 26 60)(7 72 14 59)(8 71 15 58)(9 70 16 57)(10 69 17 56)(11 68 18 55)(12 67 19 54)(13 66 20 53)(27 101 41 88)(28 100 42 87)(29 99 43 86)(30 98 44 85)(31 97 45 84)(32 96 46 83)(33 95 47 82)(34 94 48 81)(35 93 49 80)(36 92 50 79)(37 104 51 91)(38 103 52 90)(39 102 40 89)

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

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

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

D4.D13 is a maximal subgroup of   D8⋊D13  D83D13  SD16×D13  D4.D26  D526C22  C52.C23  D4.9D26
D4.D13 is a maximal quotient of   C52.Q8  C26.Q16  D4⋊Dic13

37 conjugacy classes

class 1 2A2B4A4B8A8B13A···13F26A···26F26G···26R52A···52F
order122448813···1326···2626···2652···52
size11425226262···22···24···44···4

37 irreducible representations

dim1111222224
type+++++++-
imageC1C2C2C2D4SD16D13D26C13⋊D4D4.D13
kernelD4.D13C132C8Dic26D4×C13C26C13D4C4C2C1
# reps11111266126

Matrix representation of D4.D13 in GL4(𝔽313) generated by

1000
0100
001311
001312
,
1000
0100
001311
000312
,
0100
3122400
0010
0001
,
8116600
23223200
000183
002480
G:=sub<GL(4,GF(313))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,311,312],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,311,312],[0,312,0,0,1,24,0,0,0,0,1,0,0,0,0,1],[81,232,0,0,166,232,0,0,0,0,0,248,0,0,183,0] >;

D4.D13 in GAP, Magma, Sage, TeX

D_4.D_{13}
% in TeX

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

G:=SmallGroup(208,16);
// by ID

G=gap.SmallGroup(208,16);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-13,40,61,182,97,42,4804]);
// Polycyclic

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

Export

Subgroup lattice of D4.D13 in TeX

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