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G = C172C8order 136 = 23·17

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

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

Aliases: C172C8, C34.C4, Dic17.2C2, C2.(C17⋊C4), SmallGroup(136,3)

Series: Derived Chief Lower central Upper central

C1C17 — C172C8
C1C17C34Dic17 — C172C8
C17 — C172C8
C1C2

Generators and relations for C172C8
 G = < a,b | a17=b8=1, bab-1=a4 >

17C4
17C8

Character table of C172C8

 class 124A4B8A8B8C8D17A17B17C17D34A34B34C34D
 size 1117171717171744444444
ρ11111111111111111    trivial
ρ21111-1-1-1-111111111    linear of order 2
ρ311-1-1-ii-ii11111111    linear of order 4
ρ411-1-1i-ii-i11111111    linear of order 4
ρ51-1-iiζ8ζ83ζ85ζ871111-1-1-1-1    linear of order 8
ρ61-1i-iζ87ζ85ζ83ζ81111-1-1-1-1    linear of order 8
ρ71-1i-iζ83ζ8ζ87ζ851111-1-1-1-1    linear of order 8
ρ81-1-iiζ85ζ87ζ8ζ831111-1-1-1-1    linear of order 8
ρ944000000ζ17111710177176ζ17141712175173ζ1716171317417ζ1715179178172ζ1716171317417ζ1715179178172ζ17111710177176ζ17141712175173    orthogonal lifted from C17⋊C4
ρ1044000000ζ1715179178172ζ1716171317417ζ17111710177176ζ17141712175173ζ17111710177176ζ17141712175173ζ1715179178172ζ1716171317417    orthogonal lifted from C17⋊C4
ρ1144000000ζ17141712175173ζ17111710177176ζ1715179178172ζ1716171317417ζ1715179178172ζ1716171317417ζ17141712175173ζ17111710177176    orthogonal lifted from C17⋊C4
ρ1244000000ζ1716171317417ζ1715179178172ζ17141712175173ζ17111710177176ζ17141712175173ζ17111710177176ζ1716171317417ζ1715179178172    orthogonal lifted from C17⋊C4
ρ134-4000000ζ17141712175173ζ17111710177176ζ1715179178172ζ1716171317417171517917817217161713174171714171217517317111710177176    symplectic faithful, Schur index 2
ρ144-4000000ζ17111710177176ζ17141712175173ζ1716171317417ζ1715179178172171617131741717151791781721711171017717617141712175173    symplectic faithful, Schur index 2
ρ154-4000000ζ1715179178172ζ1716171317417ζ17111710177176ζ17141712175173171117101771761714171217517317151791781721716171317417    symplectic faithful, Schur index 2
ρ164-4000000ζ1716171317417ζ1715179178172ζ17141712175173ζ17111710177176171417121751731711171017717617161713174171715179178172    symplectic faithful, Schur index 2

Smallest permutation representation of C172C8
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)
(1 125 52 86 30 103 35 83)(2 121 68 90 31 116 51 70)(3 134 67 94 32 112 50 74)(4 130 66 98 33 108 49 78)(5 126 65 102 34 104 48 82)(6 122 64 89 18 117 47 69)(7 135 63 93 19 113 46 73)(8 131 62 97 20 109 45 77)(9 127 61 101 21 105 44 81)(10 123 60 88 22 118 43 85)(11 136 59 92 23 114 42 72)(12 132 58 96 24 110 41 76)(13 128 57 100 25 106 40 80)(14 124 56 87 26 119 39 84)(15 120 55 91 27 115 38 71)(16 133 54 95 28 111 37 75)(17 129 53 99 29 107 36 79)

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

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

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

C172C8 is a maximal subgroup of   C34.C8  C68.C4  D34.4C4  C17⋊M4(2)  C513C8
C172C8 is a maximal quotient of   C173C16  C513C8

Matrix representation of C172C8 in GL5(𝔽137)

10000
00100
00010
00001
0136527252
,
100000
01177322106
027210376
094332492
0324557131

G:=sub<GL(5,GF(137))| [1,0,0,0,0,0,0,0,0,136,0,1,0,0,52,0,0,1,0,72,0,0,0,1,52],[10,0,0,0,0,0,117,27,94,32,0,73,2,33,45,0,22,103,24,57,0,106,76,92,131] >;

C172C8 in GAP, Magma, Sage, TeX

C_{17}\rtimes_2C_8
% in TeX

G:=Group("C17:2C8");
// GroupNames label

G:=SmallGroup(136,3);
// by ID

G=gap.SmallGroup(136,3);
# by ID

G:=PCGroup([4,-2,-2,-2,-17,8,21,1667,1031]);
// Polycyclic

G:=Group<a,b|a^17=b^8=1,b*a*b^-1=a^4>;
// generators/relations

Export

Subgroup lattice of C172C8 in TeX
Character table of C172C8 in TeX

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