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

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

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

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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