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G = C173C8order 136 = 23·17

The semidirect product of C17 and C8 acting via C8/C4=C2

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C173C8, C34.2C4, C68.2C2, C4.2D17, C2.Dic17, SmallGroup(136,1)

Series: Derived Chief Lower central Upper central

C1C17 — C173C8
C1C17C34C68 — C173C8
C17 — C173C8
C1C4

Generators and relations for C173C8
 G = < a,b | a17=b8=1, bab-1=a-1 >

17C8

Smallest permutation representation of C173C8
Regular action on 136 points
Generators in S136
(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)(103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119)(120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136)
(1 120 57 86 23 103 41 69)(2 136 58 102 24 119 42 85)(3 135 59 101 25 118 43 84)(4 134 60 100 26 117 44 83)(5 133 61 99 27 116 45 82)(6 132 62 98 28 115 46 81)(7 131 63 97 29 114 47 80)(8 130 64 96 30 113 48 79)(9 129 65 95 31 112 49 78)(10 128 66 94 32 111 50 77)(11 127 67 93 33 110 51 76)(12 126 68 92 34 109 35 75)(13 125 52 91 18 108 36 74)(14 124 53 90 19 107 37 73)(15 123 54 89 20 106 38 72)(16 122 55 88 21 105 39 71)(17 121 56 87 22 104 40 70)

G:=sub<Sym(136)| (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)(103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119)(120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136), (1,120,57,86,23,103,41,69)(2,136,58,102,24,119,42,85)(3,135,59,101,25,118,43,84)(4,134,60,100,26,117,44,83)(5,133,61,99,27,116,45,82)(6,132,62,98,28,115,46,81)(7,131,63,97,29,114,47,80)(8,130,64,96,30,113,48,79)(9,129,65,95,31,112,49,78)(10,128,66,94,32,111,50,77)(11,127,67,93,33,110,51,76)(12,126,68,92,34,109,35,75)(13,125,52,91,18,108,36,74)(14,124,53,90,19,107,37,73)(15,123,54,89,20,106,38,72)(16,122,55,88,21,105,39,71)(17,121,56,87,22,104,40,70)>;

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)(103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119)(120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136), (1,120,57,86,23,103,41,69)(2,136,58,102,24,119,42,85)(3,135,59,101,25,118,43,84)(4,134,60,100,26,117,44,83)(5,133,61,99,27,116,45,82)(6,132,62,98,28,115,46,81)(7,131,63,97,29,114,47,80)(8,130,64,96,30,113,48,79)(9,129,65,95,31,112,49,78)(10,128,66,94,32,111,50,77)(11,127,67,93,33,110,51,76)(12,126,68,92,34,109,35,75)(13,125,52,91,18,108,36,74)(14,124,53,90,19,107,37,73)(15,123,54,89,20,106,38,72)(16,122,55,88,21,105,39,71)(17,121,56,87,22,104,40,70) );

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),(103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119),(120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136)], [(1,120,57,86,23,103,41,69),(2,136,58,102,24,119,42,85),(3,135,59,101,25,118,43,84),(4,134,60,100,26,117,44,83),(5,133,61,99,27,116,45,82),(6,132,62,98,28,115,46,81),(7,131,63,97,29,114,47,80),(8,130,64,96,30,113,48,79),(9,129,65,95,31,112,49,78),(10,128,66,94,32,111,50,77),(11,127,67,93,33,110,51,76),(12,126,68,92,34,109,35,75),(13,125,52,91,18,108,36,74),(14,124,53,90,19,107,37,73),(15,123,54,89,20,106,38,72),(16,122,55,88,21,105,39,71),(17,121,56,87,22,104,40,70)])

C173C8 is a maximal subgroup of
C173C16  C8×D17  C8⋊D17  C68.4C4  D4⋊D17  D4.D17  Q8⋊D17  C17⋊Q16  C515C8
C173C8 is a maximal quotient of
C174C16  C515C8

40 conjugacy classes

class 1  2 4A4B8A8B8C8D17A···17H34A···34H68A···68P
order1244888817···1734···3468···68
size1111171717172···22···22···2

40 irreducible representations

dim1111222
type+++-
imageC1C2C4C8D17Dic17C173C8
kernelC173C8C68C34C17C4C2C1
# reps11248816

Matrix representation of C173C8 in GL2(𝔽137) generated by

01
13630
,
9916
10938
G:=sub<GL(2,GF(137))| [0,136,1,30],[99,109,16,38] >;

C173C8 in GAP, Magma, Sage, TeX

C_{17}\rtimes_3C_8
% in TeX

G:=Group("C17:3C8");
// GroupNames label

G:=SmallGroup(136,1);
// by ID

G=gap.SmallGroup(136,1);
# by ID

G:=PCGroup([4,-2,-2,-2,-17,8,21,2051]);
// Polycyclic

G:=Group<a,b|a^17=b^8=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C173C8 in TeX

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