C52.4C4 - 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)

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

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

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

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

C₅₂.₄C₄ is a maximal subgroup of
D₅₂⋊₄C₄ C₁₀₄.₆C₄ C₅₂.₅₃D₄ C₅₂.₄₆D₄ C₄.₁₂D₅₂ C₅₂.D₄ C₅₂.₁₀D₄ C₅₂.₅₆D₄ D₅₂.₃C₄ M₄(2)×D₁₃ D₅₂⋊₆C₂² Q₈.D₂₆ D₄.Dic₁₃ D₄⋊D₂₆ D₄.₉D₂₆
C₅₂.₄C₄ is a maximal quotient of
C₂₆.₇C₄² C₅₂⋊₃C₈ C₅₂.₅₅D₄

58 conjugacy classes

class	1	2A	2B	4A	4B	4C	8A	8B	8C	8D	13A	···	13F	26A	···	26R	52A	···	52X
order	1	2	2	4	4	4	8	8	8	8	13	···	13	26	···	26	52	···	52
size	1	1	2	1	1	2	26	26	26	26	2	···	2	2	···	2	2	···	2

58 irreducible representations

dim	1	1	1	1	1	2	2	2	2	2	2
type	+	+	+				+	-	+	-
image	C₁	C₂	C₂	C₄	C₄	M₄(2)	D₁₃	Dic₁₃	D₂₆	Dic₁₃	C₅₂.₄C₄
kernel	C₅₂.₄C₄	C₁₃⋊₂C₈	C₂×C₅₂	C₅₂	C₂×C₂₆	C₁₃	C₂×C₄	C₄	C₄	C₂²	C₁
# reps	1	2	1	2	2	2	6	6	6	6	24

Matrix representation of C₅₂.₄C₄ ►in GL₂(𝔽₃₁₃) generated by

162	0
74	114

32	30
205	281

G:=sub<GL(2,GF(313))| [162,74,0,114],[32,205,30,281] >;

C₅₂.₄C₄ in GAP, Magma, Sage, TeX

C_{52}._4C_4

% in TeX

G:=Group("C52.4C4");

// GroupNames label

G:=SmallGroup(208,10);

// by ID

G=gap.SmallGroup(208,10);

# by ID

G:=PCGroup([5,-2,-2,-2,-2,-13,20,101,42,4804]);

// Polycyclic

G:=Group<a,b|a^52=1,b^4=a^26,b*a*b^-1=a^-1>;

// generators/relations

Export

Subgroup lattice of C₅₂.₄C₄ in TeX

G = C52.4C4 order 208 = 24·13

1st non-split extension by C52 of C4 acting via C4/C2=C2

G = C₅₂.₄C₄ order 208 = 2⁴·13

1^st non-split extension by C₅₂ of C₄ acting via C₄/C₂=C₂