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G = D232order 464 = 24·29

Dihedral group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: D232, C291D8, C81D29, C2321C2, C4.9D58, C58.2D4, D1161C2, C2.4D116, C116.9C22, sometimes denoted D464 or Dih232 or Dih464, SmallGroup(464,7)

Series: Derived Chief Lower central Upper central

C1C116 — D232
C1C29C58C116D116 — D232
C29C58C116 — D232
C1C2C4C8

Generators and relations for D232
 G = < a,b | a232=b2=1, bab=a-1 >

116C2
116C2
58C22
58C22
4D29
4D29
29D4
29D4
2D58
2D58
29D8

Smallest permutation representation of D232
On 232 points
Generators in S232
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232)
(1 87)(2 86)(3 85)(4 84)(5 83)(6 82)(7 81)(8 80)(9 79)(10 78)(11 77)(12 76)(13 75)(14 74)(15 73)(16 72)(17 71)(18 70)(19 69)(20 68)(21 67)(22 66)(23 65)(24 64)(25 63)(26 62)(27 61)(28 60)(29 59)(30 58)(31 57)(32 56)(33 55)(34 54)(35 53)(36 52)(37 51)(38 50)(39 49)(40 48)(41 47)(42 46)(43 45)(88 232)(89 231)(90 230)(91 229)(92 228)(93 227)(94 226)(95 225)(96 224)(97 223)(98 222)(99 221)(100 220)(101 219)(102 218)(103 217)(104 216)(105 215)(106 214)(107 213)(108 212)(109 211)(110 210)(111 209)(112 208)(113 207)(114 206)(115 205)(116 204)(117 203)(118 202)(119 201)(120 200)(121 199)(122 198)(123 197)(124 196)(125 195)(126 194)(127 193)(128 192)(129 191)(130 190)(131 189)(132 188)(133 187)(134 186)(135 185)(136 184)(137 183)(138 182)(139 181)(140 180)(141 179)(142 178)(143 177)(144 176)(145 175)(146 174)(147 173)(148 172)(149 171)(150 170)(151 169)(152 168)(153 167)(154 166)(155 165)(156 164)(157 163)(158 162)(159 161)

G:=sub<Sym(232)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232), (1,87)(2,86)(3,85)(4,84)(5,83)(6,82)(7,81)(8,80)(9,79)(10,78)(11,77)(12,76)(13,75)(14,74)(15,73)(16,72)(17,71)(18,70)(19,69)(20,68)(21,67)(22,66)(23,65)(24,64)(25,63)(26,62)(27,61)(28,60)(29,59)(30,58)(31,57)(32,56)(33,55)(34,54)(35,53)(36,52)(37,51)(38,50)(39,49)(40,48)(41,47)(42,46)(43,45)(88,232)(89,231)(90,230)(91,229)(92,228)(93,227)(94,226)(95,225)(96,224)(97,223)(98,222)(99,221)(100,220)(101,219)(102,218)(103,217)(104,216)(105,215)(106,214)(107,213)(108,212)(109,211)(110,210)(111,209)(112,208)(113,207)(114,206)(115,205)(116,204)(117,203)(118,202)(119,201)(120,200)(121,199)(122,198)(123,197)(124,196)(125,195)(126,194)(127,193)(128,192)(129,191)(130,190)(131,189)(132,188)(133,187)(134,186)(135,185)(136,184)(137,183)(138,182)(139,181)(140,180)(141,179)(142,178)(143,177)(144,176)(145,175)(146,174)(147,173)(148,172)(149,171)(150,170)(151,169)(152,168)(153,167)(154,166)(155,165)(156,164)(157,163)(158,162)(159,161)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232), (1,87)(2,86)(3,85)(4,84)(5,83)(6,82)(7,81)(8,80)(9,79)(10,78)(11,77)(12,76)(13,75)(14,74)(15,73)(16,72)(17,71)(18,70)(19,69)(20,68)(21,67)(22,66)(23,65)(24,64)(25,63)(26,62)(27,61)(28,60)(29,59)(30,58)(31,57)(32,56)(33,55)(34,54)(35,53)(36,52)(37,51)(38,50)(39,49)(40,48)(41,47)(42,46)(43,45)(88,232)(89,231)(90,230)(91,229)(92,228)(93,227)(94,226)(95,225)(96,224)(97,223)(98,222)(99,221)(100,220)(101,219)(102,218)(103,217)(104,216)(105,215)(106,214)(107,213)(108,212)(109,211)(110,210)(111,209)(112,208)(113,207)(114,206)(115,205)(116,204)(117,203)(118,202)(119,201)(120,200)(121,199)(122,198)(123,197)(124,196)(125,195)(126,194)(127,193)(128,192)(129,191)(130,190)(131,189)(132,188)(133,187)(134,186)(135,185)(136,184)(137,183)(138,182)(139,181)(140,180)(141,179)(142,178)(143,177)(144,176)(145,175)(146,174)(147,173)(148,172)(149,171)(150,170)(151,169)(152,168)(153,167)(154,166)(155,165)(156,164)(157,163)(158,162)(159,161) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232)], [(1,87),(2,86),(3,85),(4,84),(5,83),(6,82),(7,81),(8,80),(9,79),(10,78),(11,77),(12,76),(13,75),(14,74),(15,73),(16,72),(17,71),(18,70),(19,69),(20,68),(21,67),(22,66),(23,65),(24,64),(25,63),(26,62),(27,61),(28,60),(29,59),(30,58),(31,57),(32,56),(33,55),(34,54),(35,53),(36,52),(37,51),(38,50),(39,49),(40,48),(41,47),(42,46),(43,45),(88,232),(89,231),(90,230),(91,229),(92,228),(93,227),(94,226),(95,225),(96,224),(97,223),(98,222),(99,221),(100,220),(101,219),(102,218),(103,217),(104,216),(105,215),(106,214),(107,213),(108,212),(109,211),(110,210),(111,209),(112,208),(113,207),(114,206),(115,205),(116,204),(117,203),(118,202),(119,201),(120,200),(121,199),(122,198),(123,197),(124,196),(125,195),(126,194),(127,193),(128,192),(129,191),(130,190),(131,189),(132,188),(133,187),(134,186),(135,185),(136,184),(137,183),(138,182),(139,181),(140,180),(141,179),(142,178),(143,177),(144,176),(145,175),(146,174),(147,173),(148,172),(149,171),(150,170),(151,169),(152,168),(153,167),(154,166),(155,165),(156,164),(157,163),(158,162),(159,161)])

119 conjugacy classes

class 1 2A2B2C 4 8A8B29A···29N58A···58N116A···116AB232A···232BD
order122248829···2958···58116···116232···232
size111161162222···22···22···22···2

119 irreducible representations

dim111222222
type+++++++++
imageC1C2C2D4D8D29D58D116D232
kernelD232C232D116C58C29C8C4C2C1
# reps1121214142856

Matrix representation of D232 in GL2(𝔽233) generated by

5587
78170
,
17112
64216
G:=sub<GL(2,GF(233))| [55,78,87,170],[17,64,112,216] >;

D232 in GAP, Magma, Sage, TeX

D_{232}
% in TeX

G:=Group("D232");
// GroupNames label

G:=SmallGroup(464,7);
// by ID

G=gap.SmallGroup(464,7);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-29,61,66,182,42,11204]);
// Polycyclic

G:=Group<a,b|a^232=b^2=1,b*a*b=a^-1>;
// generators/relations

Export

Subgroup lattice of D232 in TeX

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