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G = C104⋊C2order 208 = 24·13

2nd semidirect product of C104 and C2 acting faithfully

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C82D13, C1042C2, C2.3D52, C26.1D4, C4.8D26, C131SD16, D52.1C2, Dic261C2, C52.8C22, SmallGroup(208,6)

Series: Derived Chief Lower central Upper central

C1C52 — C104⋊C2
C1C13C26C52D52 — C104⋊C2
C13C26C52 — C104⋊C2
C1C2C4C8

Generators and relations for C104⋊C2
 G = < a,b | a104=b2=1, bab=a51 >

52C2
26C22
26C4
4D13
13Q8
13D4
2Dic13
2D26
13SD16

Smallest permutation representation of C104⋊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)
(2 52)(3 103)(4 50)(5 101)(6 48)(7 99)(8 46)(9 97)(10 44)(11 95)(12 42)(13 93)(14 40)(15 91)(16 38)(17 89)(18 36)(19 87)(20 34)(21 85)(22 32)(23 83)(24 30)(25 81)(26 28)(27 79)(29 77)(31 75)(33 73)(35 71)(37 69)(39 67)(41 65)(43 63)(45 61)(47 59)(49 57)(51 55)(54 104)(56 102)(58 100)(60 98)(62 96)(64 94)(66 92)(68 90)(70 88)(72 86)(74 84)(76 82)(78 80)

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

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

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

C104⋊C2 is a maximal subgroup of   D1047C2  C8⋊D26  C8.D26  D8⋊D13  SD16×D13  D26.6D4  Q16⋊D13
C104⋊C2 is a maximal quotient of   C52.44D4  C1046C4  D525C4

55 conjugacy classes

class 1 2A2B4A4B8A8B13A···13F26A···26F52A···52L104A···104X
order122448813···1326···2652···52104···104
size1152252222···22···22···22···2

55 irreducible representations

dim1111222222
type++++++++
imageC1C2C2C2D4SD16D13D26D52C104⋊C2
kernelC104⋊C2C104Dic26D52C26C13C8C4C2C1
# reps111112661224

Matrix representation of C104⋊C2 in GL2(𝔽313) generated by

17436
25266
,
135115
13178
G:=sub<GL(2,GF(313))| [174,25,36,266],[135,13,115,178] >;

C104⋊C2 in GAP, Magma, Sage, TeX

C_{104}\rtimes C_2
% in TeX

G:=Group("C104:C2");
// GroupNames label

G:=SmallGroup(208,6);
// by ID

G=gap.SmallGroup(208,6);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-13,61,26,182,42,4804]);
// Polycyclic

G:=Group<a,b|a^104=b^2=1,b*a*b=a^51>;
// generators/relations

Export

Subgroup lattice of C104⋊C2 in TeX

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