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## G = C16⋊5C8order 128 = 27

### 3rd semidirect product of C16 and C8 acting via C8/C4=C2

p-group, metacyclic, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C16⋊5C8
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C4×C8 — C82 — C16⋊5C8
 Lower central C1 — C2 — C16⋊5C8
 Upper central C1 — C4×C8 — C16⋊5C8
 Jennings C1 — C2 — C2 — C2 — C2 — C2×C4 — C2×C4 — C4×C8 — C16⋊5C8

Generators and relations for C165C8
G = < a,b | a16=b8=1, bab-1=a9 >

Smallest permutation representation of C165C8
Regular action on 128 points
Generators in S128
(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)
(1 22 34 86 58 128 102 73)(2 31 35 95 59 121 103 66)(3 24 36 88 60 114 104 75)(4 17 37 81 61 123 105 68)(5 26 38 90 62 116 106 77)(6 19 39 83 63 125 107 70)(7 28 40 92 64 118 108 79)(8 21 41 85 49 127 109 72)(9 30 42 94 50 120 110 65)(10 23 43 87 51 113 111 74)(11 32 44 96 52 122 112 67)(12 25 45 89 53 115 97 76)(13 18 46 82 54 124 98 69)(14 27 47 91 55 117 99 78)(15 20 48 84 56 126 100 71)(16 29 33 93 57 119 101 80)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4L 8A ··· 8P 8Q ··· 8AF 16A ··· 16AF order 1 2 2 2 4 ··· 4 8 ··· 8 8 ··· 8 16 ··· 16 size 1 1 1 1 1 ··· 1 1 ··· 1 2 ··· 2 2 ··· 2

80 irreducible representations

 dim 1 1 1 1 1 1 1 2 type + + + image C1 C2 C2 C4 C4 C8 C8 M5(2) kernel C16⋊5C8 C82 C4×C16 C4×C8 C2×C16 C16 C2×C8 C4 # reps 1 1 2 4 8 32 16 16

Matrix representation of C165C8 in GL3(𝔽17) generated by

 13 0 0 0 10 16 0 13 7
,
 9 0 0 0 0 1 0 13 0
G:=sub<GL(3,GF(17))| [13,0,0,0,10,13,0,16,7],[9,0,0,0,0,13,0,1,0] >;

C165C8 in GAP, Magma, Sage, TeX

C_{16}\rtimes_5C_8
% in TeX

G:=Group("C16:5C8");
// GroupNames label

G:=SmallGroup(128,43);
// by ID

G=gap.SmallGroup(128,43);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,28,925,64,100,136,172]);
// Polycyclic

G:=Group<a,b|a^16=b^8=1,b*a*b^-1=a^9>;
// generators/relations

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