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G = C165C8order 128 = 27

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

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

Aliases: C165C8, C2.2C82, C82.1C2, C4.9M5(2), (C2×C8).5C8, C8.27(C2×C8), C4.11(C4×C8), (C4×C8).12C4, (C2×C16).16C4, (C4×C16).15C2, (C2×C4).80C42, C22.12(C4×C8), C2.1(C165C4), C42.335(C2×C4), (C4×C8).439C22, (C2×C4).92(C2×C8), (C2×C8).256(C2×C4), SmallGroup(128,43)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C165C8
C1C2C22C2×C4C42C4×C8C82 — C165C8
C1C2 — C165C8
C1C4×C8 — C165C8
C1C2C2C2C2C2×C4C2×C4C4×C8 — C165C8

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

2C8
2C8
2C8
2C8

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

80 irreducible representations

dim11111112
type+++
imageC1C2C2C4C4C8C8M5(2)
kernelC165C8C82C4×C16C4×C8C2×C16C16C2×C8C4
# reps11248321616

Matrix representation of C165C8 in GL3(𝔽17) generated by

1300
01016
0137
,
900
001
0130
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

Export

Subgroup lattice of C165C8 in TeX

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