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## G = C104⋊C2order 208 = 24·13

### 2nd semidirect product of C104 and C2 acting faithfully

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

 Derived series C1 — C52 — C104⋊C2
 Chief series C1 — C13 — C26 — C52 — D52 — C104⋊C2
 Lower central C13 — C26 — C52 — C104⋊C2
 Upper central C1 — C2 — C4 — C8

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

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 2A 2B 4A 4B 8A 8B 13A ··· 13F 26A ··· 26F 52A ··· 52L 104A ··· 104X order 1 2 2 4 4 8 8 13 ··· 13 26 ··· 26 52 ··· 52 104 ··· 104 size 1 1 52 2 52 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

55 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 D4 SD16 D13 D26 D52 C104⋊C2 kernel C104⋊C2 C104 Dic26 D52 C26 C13 C8 C4 C2 C1 # reps 1 1 1 1 1 2 6 6 12 24

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

 174 36 25 266
,
 135 115 13 178
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

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