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G = C3×C84order 252 = 22·32·7

Abelian group of type [3,84]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C84, SmallGroup(252,25)

Series: Derived Chief Lower central Upper central

C1 — C3×C84
C1C2C14C42C3×C42 — C3×C84
C1 — C3×C84
C1 — C3×C84

Generators and relations for C3×C84
 G = < a,b | a3=b84=1, ab=ba >


Smallest permutation representation of C3×C84
Regular action on 252 points
Generators in S252
(1 128 190)(2 129 191)(3 130 192)(4 131 193)(5 132 194)(6 133 195)(7 134 196)(8 135 197)(9 136 198)(10 137 199)(11 138 200)(12 139 201)(13 140 202)(14 141 203)(15 142 204)(16 143 205)(17 144 206)(18 145 207)(19 146 208)(20 147 209)(21 148 210)(22 149 211)(23 150 212)(24 151 213)(25 152 214)(26 153 215)(27 154 216)(28 155 217)(29 156 218)(30 157 219)(31 158 220)(32 159 221)(33 160 222)(34 161 223)(35 162 224)(36 163 225)(37 164 226)(38 165 227)(39 166 228)(40 167 229)(41 168 230)(42 85 231)(43 86 232)(44 87 233)(45 88 234)(46 89 235)(47 90 236)(48 91 237)(49 92 238)(50 93 239)(51 94 240)(52 95 241)(53 96 242)(54 97 243)(55 98 244)(56 99 245)(57 100 246)(58 101 247)(59 102 248)(60 103 249)(61 104 250)(62 105 251)(63 106 252)(64 107 169)(65 108 170)(66 109 171)(67 110 172)(68 111 173)(69 112 174)(70 113 175)(71 114 176)(72 115 177)(73 116 178)(74 117 179)(75 118 180)(76 119 181)(77 120 182)(78 121 183)(79 122 184)(80 123 185)(81 124 186)(82 125 187)(83 126 188)(84 127 189)
(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)(169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252)

G:=sub<Sym(252)| (1,128,190)(2,129,191)(3,130,192)(4,131,193)(5,132,194)(6,133,195)(7,134,196)(8,135,197)(9,136,198)(10,137,199)(11,138,200)(12,139,201)(13,140,202)(14,141,203)(15,142,204)(16,143,205)(17,144,206)(18,145,207)(19,146,208)(20,147,209)(21,148,210)(22,149,211)(23,150,212)(24,151,213)(25,152,214)(26,153,215)(27,154,216)(28,155,217)(29,156,218)(30,157,219)(31,158,220)(32,159,221)(33,160,222)(34,161,223)(35,162,224)(36,163,225)(37,164,226)(38,165,227)(39,166,228)(40,167,229)(41,168,230)(42,85,231)(43,86,232)(44,87,233)(45,88,234)(46,89,235)(47,90,236)(48,91,237)(49,92,238)(50,93,239)(51,94,240)(52,95,241)(53,96,242)(54,97,243)(55,98,244)(56,99,245)(57,100,246)(58,101,247)(59,102,248)(60,103,249)(61,104,250)(62,105,251)(63,106,252)(64,107,169)(65,108,170)(66,109,171)(67,110,172)(68,111,173)(69,112,174)(70,113,175)(71,114,176)(72,115,177)(73,116,178)(74,117,179)(75,118,180)(76,119,181)(77,120,182)(78,121,183)(79,122,184)(80,123,185)(81,124,186)(82,125,187)(83,126,188)(84,127,189), (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)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252)>;

G:=Group( (1,128,190)(2,129,191)(3,130,192)(4,131,193)(5,132,194)(6,133,195)(7,134,196)(8,135,197)(9,136,198)(10,137,199)(11,138,200)(12,139,201)(13,140,202)(14,141,203)(15,142,204)(16,143,205)(17,144,206)(18,145,207)(19,146,208)(20,147,209)(21,148,210)(22,149,211)(23,150,212)(24,151,213)(25,152,214)(26,153,215)(27,154,216)(28,155,217)(29,156,218)(30,157,219)(31,158,220)(32,159,221)(33,160,222)(34,161,223)(35,162,224)(36,163,225)(37,164,226)(38,165,227)(39,166,228)(40,167,229)(41,168,230)(42,85,231)(43,86,232)(44,87,233)(45,88,234)(46,89,235)(47,90,236)(48,91,237)(49,92,238)(50,93,239)(51,94,240)(52,95,241)(53,96,242)(54,97,243)(55,98,244)(56,99,245)(57,100,246)(58,101,247)(59,102,248)(60,103,249)(61,104,250)(62,105,251)(63,106,252)(64,107,169)(65,108,170)(66,109,171)(67,110,172)(68,111,173)(69,112,174)(70,113,175)(71,114,176)(72,115,177)(73,116,178)(74,117,179)(75,118,180)(76,119,181)(77,120,182)(78,121,183)(79,122,184)(80,123,185)(81,124,186)(82,125,187)(83,126,188)(84,127,189), (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)(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252) );

G=PermutationGroup([(1,128,190),(2,129,191),(3,130,192),(4,131,193),(5,132,194),(6,133,195),(7,134,196),(8,135,197),(9,136,198),(10,137,199),(11,138,200),(12,139,201),(13,140,202),(14,141,203),(15,142,204),(16,143,205),(17,144,206),(18,145,207),(19,146,208),(20,147,209),(21,148,210),(22,149,211),(23,150,212),(24,151,213),(25,152,214),(26,153,215),(27,154,216),(28,155,217),(29,156,218),(30,157,219),(31,158,220),(32,159,221),(33,160,222),(34,161,223),(35,162,224),(36,163,225),(37,164,226),(38,165,227),(39,166,228),(40,167,229),(41,168,230),(42,85,231),(43,86,232),(44,87,233),(45,88,234),(46,89,235),(47,90,236),(48,91,237),(49,92,238),(50,93,239),(51,94,240),(52,95,241),(53,96,242),(54,97,243),(55,98,244),(56,99,245),(57,100,246),(58,101,247),(59,102,248),(60,103,249),(61,104,250),(62,105,251),(63,106,252),(64,107,169),(65,108,170),(66,109,171),(67,110,172),(68,111,173),(69,112,174),(70,113,175),(71,114,176),(72,115,177),(73,116,178),(74,117,179),(75,118,180),(76,119,181),(77,120,182),(78,121,183),(79,122,184),(80,123,185),(81,124,186),(82,125,187),(83,126,188),(84,127,189)], [(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),(169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252)])

252 conjugacy classes

class 1  2 3A···3H4A4B6A···6H7A···7F12A···12P14A···14F21A···21AV28A···28L42A···42AV84A···84CR
order123···3446···67···712···1214···1421···2128···2842···4284···84
size111···1111···11···11···11···11···11···11···11···1

252 irreducible representations

dim111111111111
type++
imageC1C2C3C4C6C7C12C14C21C28C42C84
kernelC3×C84C3×C42C84C3×C21C42C3×C12C21C3×C6C12C32C6C3
# reps11828616648124896

Matrix representation of C3×C84 in GL2(𝔽337) generated by

2080
01
,
80
09
G:=sub<GL(2,GF(337))| [208,0,0,1],[8,0,0,9] >;

C3×C84 in GAP, Magma, Sage, TeX

C_3\times C_{84}
% in TeX

G:=Group("C3xC84");
// GroupNames label

G:=SmallGroup(252,25);
// by ID

G=gap.SmallGroup(252,25);
# by ID

G:=PCGroup([5,-2,-3,-3,-7,-2,630]);
// Polycyclic

G:=Group<a,b|a^3=b^84=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C3×C84 in TeX

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