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G = C16⋊C8order 128 = 27

2nd semidirect product of C16 and C8 acting via C8/C2=C4

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

Aliases: C162C8, C8.20M4(2), (C4×C8).3C4, C4.18(C4×C8), C8.21(C2×C8), (C2×C16).3C4, C8⋊C8.8C2, C165C4.5C2, C2.3(C8⋊C8), (C2×C4).82C42, C2.1(C16⋊C4), C4.14(C8⋊C4), (C4×C8).302C22, C42.291(C2×C4), (C2×C4).53M4(2), C22.11(C8⋊C4), (C2×C8).236(C2×C4), SmallGroup(128,45)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — C16⋊C8
C1C2C22C2×C4C42C4×C8C8⋊C8 — C16⋊C8
C1C4 — C16⋊C8
C1C2×C4 — C16⋊C8
C1C2C2C2C2C2×C4C2×C4C4×C8 — C16⋊C8

Generators and relations for C16⋊C8
 G = < a,b | a16=b8=1, bab-1=a5 >

2C4
2C4
4C8
4C8
2C2×C8
2C2×C8
2C16
2C16

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

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

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

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

44 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H8A···8H8I···8P16A···16P
order1222444444448···88···816···16
size1111111122222···24···44···4

44 irreducible representations

dim111111224
type+++
imageC1C2C2C4C4C8M4(2)M4(2)C16⋊C4
kernelC16⋊C8C8⋊C8C165C4C4×C8C2×C16C16C8C2×C4C2
# reps1124816444

Matrix representation of C16⋊C8 in GL6(𝔽17)

3100000
14140000
0061520
0000416
001471516
00157913
,
880000
190000
0091600
0014800
001513615
0058911

G:=sub<GL(6,GF(17))| [3,14,0,0,0,0,10,14,0,0,0,0,0,0,6,0,14,15,0,0,15,0,7,7,0,0,2,4,15,9,0,0,0,16,16,13],[8,1,0,0,0,0,8,9,0,0,0,0,0,0,9,14,15,5,0,0,16,8,13,8,0,0,0,0,6,9,0,0,0,0,15,11] >;

C16⋊C8 in GAP, Magma, Sage, TeX

C_{16}\rtimes C_8
% in TeX

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

G:=SmallGroup(128,45);
// by ID

G=gap.SmallGroup(128,45);
# by ID

G:=PCGroup([7,-2,2,-2,2,-2,2,-2,28,253,64,723,100,2019,136,172]);
// Polycyclic

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

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Subgroup lattice of C16⋊C8 in TeX

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