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G = C8×C16order 128 = 27

Abelian group of type [8,16]

direct product, p-group, abelian, monomial

Aliases: C8×C16, SmallGroup(128,42)

Series: Derived Chief Lower central Upper central Jennings

C1 — C8×C16
C1C2C22C2×C4C42C4×C8C82 — C8×C16
C1 — C8×C16
C1 — C8×C16
C1C2C2C2C2C2×C4C2×C4C4×C8 — C8×C16

Generators and relations for C8×C16
 G = < a,b | a8=b16=1, ab=ba >


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

128 conjugacy classes

class 1 2A2B2C4A···4L8A···8AV16A···16BL
order12224···48···816···16
size11111···11···11···1

128 irreducible representations

dim11111111
type+++
imageC1C2C2C4C4C8C8C16
kernelC8×C16C82C4×C16C4×C8C2×C16C16C2×C8C8
# reps11248321664

Matrix representation of C8×C16 in GL2(𝔽17) generated by

80
04
,
140
03
G:=sub<GL(2,GF(17))| [8,0,0,4],[14,0,0,3] >;

C8×C16 in GAP, Magma, Sage, TeX

C_8\times C_{16}
% in TeX

G:=Group("C8xC16");
// GroupNames label

G:=SmallGroup(128,42);
// by ID

G=gap.SmallGroup(128,42);
# by ID

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

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

Export

Subgroup lattice of C8×C16 in TeX

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