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G = C11⋊C16order 176 = 24·11

The semidirect product of C11 and C16 acting via C16/C8=C2

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C11⋊C16, C22.C8, C44.2C4, C88.2C2, C8.2D11, C4.2Dic11, C2.(C11⋊C8), SmallGroup(176,1)

Series: Derived Chief Lower central Upper central

C1C11 — C11⋊C16
C1C11C22C44C88 — C11⋊C16
C11 — C11⋊C16
C1C8

Generators and relations for C11⋊C16
 G = < a,b | a11=b16=1, bab-1=a-1 >

11C16

Smallest permutation representation of C11⋊C16
Regular action on 176 points
Generators in S176
(1 92 28 125 158 74 104 163 143 45 54)(2 55 46 144 164 105 75 159 126 29 93)(3 94 30 127 160 76 106 165 129 47 56)(4 57 48 130 166 107 77 145 128 31 95)(5 96 32 113 146 78 108 167 131 33 58)(6 59 34 132 168 109 79 147 114 17 81)(7 82 18 115 148 80 110 169 133 35 60)(8 61 36 134 170 111 65 149 116 19 83)(9 84 20 117 150 66 112 171 135 37 62)(10 63 38 136 172 97 67 151 118 21 85)(11 86 22 119 152 68 98 173 137 39 64)(12 49 40 138 174 99 69 153 120 23 87)(13 88 24 121 154 70 100 175 139 41 50)(14 51 42 140 176 101 71 155 122 25 89)(15 90 26 123 156 72 102 161 141 43 52)(16 53 44 142 162 103 73 157 124 27 91)
(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)(129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144)(145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)(161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176)

G:=sub<Sym(176)| (1,92,28,125,158,74,104,163,143,45,54)(2,55,46,144,164,105,75,159,126,29,93)(3,94,30,127,160,76,106,165,129,47,56)(4,57,48,130,166,107,77,145,128,31,95)(5,96,32,113,146,78,108,167,131,33,58)(6,59,34,132,168,109,79,147,114,17,81)(7,82,18,115,148,80,110,169,133,35,60)(8,61,36,134,170,111,65,149,116,19,83)(9,84,20,117,150,66,112,171,135,37,62)(10,63,38,136,172,97,67,151,118,21,85)(11,86,22,119,152,68,98,173,137,39,64)(12,49,40,138,174,99,69,153,120,23,87)(13,88,24,121,154,70,100,175,139,41,50)(14,51,42,140,176,101,71,155,122,25,89)(15,90,26,123,156,72,102,161,141,43,52)(16,53,44,142,162,103,73,157,124,27,91), (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)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176)>;

G:=Group( (1,92,28,125,158,74,104,163,143,45,54)(2,55,46,144,164,105,75,159,126,29,93)(3,94,30,127,160,76,106,165,129,47,56)(4,57,48,130,166,107,77,145,128,31,95)(5,96,32,113,146,78,108,167,131,33,58)(6,59,34,132,168,109,79,147,114,17,81)(7,82,18,115,148,80,110,169,133,35,60)(8,61,36,134,170,111,65,149,116,19,83)(9,84,20,117,150,66,112,171,135,37,62)(10,63,38,136,172,97,67,151,118,21,85)(11,86,22,119,152,68,98,173,137,39,64)(12,49,40,138,174,99,69,153,120,23,87)(13,88,24,121,154,70,100,175,139,41,50)(14,51,42,140,176,101,71,155,122,25,89)(15,90,26,123,156,72,102,161,141,43,52)(16,53,44,142,162,103,73,157,124,27,91), (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)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176) );

G=PermutationGroup([(1,92,28,125,158,74,104,163,143,45,54),(2,55,46,144,164,105,75,159,126,29,93),(3,94,30,127,160,76,106,165,129,47,56),(4,57,48,130,166,107,77,145,128,31,95),(5,96,32,113,146,78,108,167,131,33,58),(6,59,34,132,168,109,79,147,114,17,81),(7,82,18,115,148,80,110,169,133,35,60),(8,61,36,134,170,111,65,149,116,19,83),(9,84,20,117,150,66,112,171,135,37,62),(10,63,38,136,172,97,67,151,118,21,85),(11,86,22,119,152,68,98,173,137,39,64),(12,49,40,138,174,99,69,153,120,23,87),(13,88,24,121,154,70,100,175,139,41,50),(14,51,42,140,176,101,71,155,122,25,89),(15,90,26,123,156,72,102,161,141,43,52),(16,53,44,142,162,103,73,157,124,27,91)], [(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),(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144),(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160),(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176)])

C11⋊C16 is a maximal subgroup of   C16×D11  D22.C8  C44.C8  C11⋊D16  D8.D11  C8.6D22  C11⋊Q32
C11⋊C16 is a maximal quotient of   C11⋊C32

56 conjugacy classes

class 1  2 4A4B8A8B8C8D11A···11E16A···16H22A···22E44A···44J88A···88T
order1244888811···1116···1622···2244···4488···88
size111111112···211···112···22···22···2

56 irreducible representations

dim111112222
type+++-
imageC1C2C4C8C16D11Dic11C11⋊C8C11⋊C16
kernelC11⋊C16C88C44C22C11C8C4C2C1
# reps11248551020

Matrix representation of C11⋊C16 in GL3(𝔽353) generated by

100
001
035295
,
29300
016339
094337
G:=sub<GL(3,GF(353))| [1,0,0,0,0,352,0,1,95],[293,0,0,0,16,94,0,339,337] >;

C11⋊C16 in GAP, Magma, Sage, TeX

C_{11}\rtimes C_{16}
% in TeX

G:=Group("C11:C16");
// GroupNames label

G:=SmallGroup(176,1);
// by ID

G=gap.SmallGroup(176,1);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-11,10,26,42,4004]);
// Polycyclic

G:=Group<a,b|a^11=b^16=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C11⋊C16 in TeX

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