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G = C2xD52order 208 = 24·13

Direct product of C2 and D52

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2xD52, C4:2D26, C26:1D4, C52:2C22, D26:1C22, C26.3C23, C22.10D26, C13:1(C2xD4), (C2xC52):3C2, (C2xC4):2D13, (C22xD13):1C2, C2.4(C22xD13), (C2xC26).10C22, SmallGroup(208,37)

Series: Derived Chief Lower central Upper central

C1C26 — C2xD52
C1C13C26D26C22xD13 — C2xD52
C13C26 — C2xD52
C1C22C2xC4

Generators and relations for C2xD52
 G = < a,b,c | a2=b52=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 394 in 54 conjugacy classes, 27 normal (9 characteristic)
C1, C2, C2, C2, C4, C22, C22, C2xC4, D4, C23, C13, C2xD4, D13, C26, C26, C52, D26, D26, C2xC26, D52, C2xC52, C22xD13, C2xD52
Quotients: C1, C2, C22, D4, C23, C2xD4, D13, D26, D52, C22xD13, C2xD52

Smallest permutation representation of C2xD52
On 104 points
Generators in S104
(1 88)(2 89)(3 90)(4 91)(5 92)(6 93)(7 94)(8 95)(9 96)(10 97)(11 98)(12 99)(13 100)(14 101)(15 102)(16 103)(17 104)(18 53)(19 54)(20 55)(21 56)(22 57)(23 58)(24 59)(25 60)(26 61)(27 62)(28 63)(29 64)(30 65)(31 66)(32 67)(33 68)(34 69)(35 70)(36 71)(37 72)(38 73)(39 74)(40 75)(41 76)(42 77)(43 78)(44 79)(45 80)(46 81)(47 82)(48 83)(49 84)(50 85)(51 86)(52 87)
(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 70)(54 69)(55 68)(56 67)(57 66)(58 65)(59 64)(60 63)(61 62)(71 104)(72 103)(73 102)(74 101)(75 100)(76 99)(77 98)(78 97)(79 96)(80 95)(81 94)(82 93)(83 92)(84 91)(85 90)(86 89)(87 88)

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

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

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

C2xD52 is a maximal subgroup of
D52:6C4  D52:5C4  C52.46D4  D26.D4  C4:D52  C4.D52  C22:D52  D26:D4  D52:8C4  D26.13D4  C4:2D52  C8:D26  C52:7D4  C52:D4  C52.23D4  D4:D26  C2xD4xD13  D4:8D26
C2xD52 is a maximal quotient of
C52:2Q8  C4:D52  C4.D52  C22:D52  C22.D52  C4:2D52  D26:2Q8  D104:7C2  C8:D26  C8.D26  C52:7D4

58 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B13A···13F26A···26R52A···52X
order122222224413···1326···2652···52
size111126262626222···22···22···2

58 irreducible representations

dim111122222
type+++++++++
imageC1C2C2C2D4D13D26D26D52
kernelC2xD52D52C2xC52C22xD13C26C2xC4C4C22C2
# reps14122612624

Matrix representation of C2xD52 in GL4(F53) generated by

52000
05200
0010
0001
,
361000
02800
003338
00045
,
15100
413800
002718
004226
G:=sub<GL(4,GF(53))| [52,0,0,0,0,52,0,0,0,0,1,0,0,0,0,1],[36,0,0,0,10,28,0,0,0,0,33,0,0,0,38,45],[15,41,0,0,1,38,0,0,0,0,27,42,0,0,18,26] >;

C2xD52 in GAP, Magma, Sage, TeX

C_2\times D_{52}
% in TeX

G:=Group("C2xD52");
// GroupNames label

G:=SmallGroup(208,37);
// by ID

G=gap.SmallGroup(208,37);
# by ID

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

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

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