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G = C8⋊C16order 128 = 27

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

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

Aliases: C83C16, C82.9C2, C8.24M4(2), C4.10M5(2), (C2×C8).6C8, C2.1(C4×C16), (C2×C16).7C4, (C4×C16).1C2, (C4×C8).13C4, C4.11(C2×C16), C2.2(C8⋊C8), (C2×C4).81C42, C22.13(C4×C8), C4.13(C8⋊C4), C2.2(C165C4), C42.336(C2×C4), (C4×C8).440C22, (C2×C4).93(C2×C8), (C2×C8).257(C2×C4), SmallGroup(128,44)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C8⋊C16
C1C2C4C2×C4C2×C8C4×C8C4×C16 — C8⋊C16
C1C2 — C8⋊C16
C1C4×C8 — C8⋊C16
C1C2C2C2C2C2×C4C2×C4C4×C8 — C8⋊C16

Generators and relations for C8⋊C16
 G = < a,b | a8=b16=1, bab-1=a5 >

2C8
2C8
2C16
2C16
2C16
2C16

Smallest permutation representation of C8⋊C16
Regular action on 128 points
Generators in S128
(1 99 25 121 47 60 89 69)(2 61 26 70 48 100 90 122)(3 101 27 123 33 62 91 71)(4 63 28 72 34 102 92 124)(5 103 29 125 35 64 93 73)(6 49 30 74 36 104 94 126)(7 105 31 127 37 50 95 75)(8 51 32 76 38 106 96 128)(9 107 17 113 39 52 81 77)(10 53 18 78 40 108 82 114)(11 109 19 115 41 54 83 79)(12 55 20 80 42 110 84 116)(13 111 21 117 43 56 85 65)(14 57 22 66 44 112 86 118)(15 97 23 119 45 58 87 67)(16 59 24 68 46 98 88 120)
(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)

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

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

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

80 conjugacy classes

class 1 2A2B2C4A···4L8A···8P8Q···8AF16A···16AF
order12224···48···88···816···16
size11111···11···12···22···2

80 irreducible representations

dim111111122
type+++
imageC1C2C2C4C4C8C16M4(2)M5(2)
kernelC8⋊C16C82C4×C16C4×C8C2×C16C2×C8C8C8C4
# reps11248163288

Matrix representation of C8⋊C16 in GL3(𝔽17) generated by

100
0013
010
,
300
0100
007
G:=sub<GL(3,GF(17))| [1,0,0,0,0,1,0,13,0],[3,0,0,0,10,0,0,0,7] >;

C8⋊C16 in GAP, Magma, Sage, TeX

C_8\rtimes C_{16}
% in TeX

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

G:=SmallGroup(128,44);
// by ID

G=gap.SmallGroup(128,44);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,28,253,64,100,136,124]);
// Polycyclic

G:=Group<a,b|a^8=b^16=1,b*a*b^-1=a^5>;
// generators/relations

Export

Subgroup lattice of C8⋊C16 in TeX

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