C104:C2 - GroupNames

(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 D₁₀₄⋊₇C₂ C₈⋊D₂₆ C₈.D₂₆ D₈⋊D₁₃ SD₁₆×D₁₃ D₂₆.₆D₄ Q₁₆⋊D₁₃
C₁₀₄⋊C₂ is a maximal quotient of C₅₂.₄₄D₄ C₁₀₄⋊₆C₄ D₅₂⋊₅C₄

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

Export

Subgroup lattice of C₁₀₄⋊C₂ in TeX

G = C104⋊C2 order 208 = 24·13

2nd semidirect product of C104 and C2 acting faithfully

G = C₁₀₄⋊C₂ order 208 = 2⁴·13

2^nd semidirect product of C₁₀₄ and C₂ acting faithfully