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G = D52⋊C2order 208 = 24·13

4th semidirect product of D52 and C2 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D524C2, Q82D13, C4.7D26, C26.8C23, C52.7C22, D26.3C22, Dic13.9C22, (C4×D13)⋊3C2, C133(C4○D4), (Q8×C13)⋊3C2, C2.9(C22×D13), SmallGroup(208,42)

Series: Derived Chief Lower central Upper central

C1C26 — D52⋊C2
C1C13C26D26C4×D13 — D52⋊C2
C13C26 — D52⋊C2
C1C2Q8

Generators and relations for D52⋊C2
 G = < a,b,c | a52=b2=c2=1, bab=a-1, cac=a25, cbc=a50b >

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

Smallest permutation representation of D52⋊C2
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 52)(2 51)(3 50)(4 49)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 36)(18 35)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)(53 94)(54 93)(55 92)(56 91)(57 90)(58 89)(59 88)(60 87)(61 86)(62 85)(63 84)(64 83)(65 82)(66 81)(67 80)(68 79)(69 78)(70 77)(71 76)(72 75)(73 74)(95 104)(96 103)(97 102)(98 101)(99 100)
(1 87)(2 60)(3 85)(4 58)(5 83)(6 56)(7 81)(8 54)(9 79)(10 104)(11 77)(12 102)(13 75)(14 100)(15 73)(16 98)(17 71)(18 96)(19 69)(20 94)(21 67)(22 92)(23 65)(24 90)(25 63)(26 88)(27 61)(28 86)(29 59)(30 84)(31 57)(32 82)(33 55)(34 80)(35 53)(36 78)(37 103)(38 76)(39 101)(40 74)(41 99)(42 72)(43 97)(44 70)(45 95)(46 68)(47 93)(48 66)(49 91)(50 64)(51 89)(52 62)

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

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

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

D52⋊C2 is a maximal subgroup of
D52⋊C4  Q8⋊D26  D26.6D4  Q16⋊D13  D104⋊C2  D52.C4  Q8.10D26  C4○D4×D13  D48D26
D52⋊C2 is a maximal quotient of
C4.Dic26  C4⋊C47D13  D528C4  D26.13D4  C42D52  C4⋊C4⋊D13  Q8×Dic13  D263Q8  C52.23D4

40 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E13A···13F26A···26F52A···52R
order122224444413···1326···2652···52
size1126262622213132···22···24···4

40 irreducible representations

dim11112224
type+++++++
imageC1C2C2C2C4○D4D13D26D52⋊C2
kernelD52⋊C2C4×D13D52Q8×C13C13Q8C4C1
# reps133126186

Matrix representation of D52⋊C2 in GL4(𝔽53) generated by

40900
445100
00522
00521
,
40900
521300
00522
0001
,
52000
40100
00237
002330
G:=sub<GL(4,GF(53))| [40,44,0,0,9,51,0,0,0,0,52,52,0,0,2,1],[40,52,0,0,9,13,0,0,0,0,52,0,0,0,2,1],[52,40,0,0,0,1,0,0,0,0,23,23,0,0,7,30] >;

D52⋊C2 in GAP, Magma, Sage, TeX

D_{52}\rtimes C_2
% in TeX

G:=Group("D52:C2");
// GroupNames label

G:=SmallGroup(208,42);
// by ID

G=gap.SmallGroup(208,42);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-13,46,182,97,42,4804]);
// Polycyclic

G:=Group<a,b,c|a^52=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^25,c*b*c=a^50*b>;
// generators/relations

Export

Subgroup lattice of D52⋊C2 in TeX

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