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G = C3×C72order 216 = 23·33

Abelian group of type [3,72]

direct product, abelian, monomial, 3-elementary

Aliases: C3×C72, SmallGroup(216,18)

Series: Derived Chief Lower central Upper central

C1 — C3×C72
C1C2C6C12C3×C12C3×C36 — C3×C72
C1 — C3×C72
C1 — C3×C72

Generators and relations for C3×C72
 G = < a,b | a3=b72=1, ab=ba >


Smallest permutation representation of C3×C72
Regular action on 216 points
Generators in S216
(1 99 187)(2 100 188)(3 101 189)(4 102 190)(5 103 191)(6 104 192)(7 105 193)(8 106 194)(9 107 195)(10 108 196)(11 109 197)(12 110 198)(13 111 199)(14 112 200)(15 113 201)(16 114 202)(17 115 203)(18 116 204)(19 117 205)(20 118 206)(21 119 207)(22 120 208)(23 121 209)(24 122 210)(25 123 211)(26 124 212)(27 125 213)(28 126 214)(29 127 215)(30 128 216)(31 129 145)(32 130 146)(33 131 147)(34 132 148)(35 133 149)(36 134 150)(37 135 151)(38 136 152)(39 137 153)(40 138 154)(41 139 155)(42 140 156)(43 141 157)(44 142 158)(45 143 159)(46 144 160)(47 73 161)(48 74 162)(49 75 163)(50 76 164)(51 77 165)(52 78 166)(53 79 167)(54 80 168)(55 81 169)(56 82 170)(57 83 171)(58 84 172)(59 85 173)(60 86 174)(61 87 175)(62 88 176)(63 89 177)(64 90 178)(65 91 179)(66 92 180)(67 93 181)(68 94 182)(69 95 183)(70 96 184)(71 97 185)(72 98 186)
(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)

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

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

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

C3×C72 is a maximal subgroup of   C72.S3  C24.D9  C72⋊S3  C24⋊D9  C721S3

216 conjugacy classes

class 1  2 3A···3H4A4B6A···6H8A8B8C8D9A···9R12A···12P18A···18R24A···24AF36A···36AJ72A···72BT
order123···3446···688889···912···1218···1824···2436···3672···72
size111···1111···111111···11···11···11···11···11···1

216 irreducible representations

dim1111111111111111
type++
imageC1C2C3C3C4C6C6C8C9C12C12C18C24C24C36C72
kernelC3×C72C3×C36C72C3×C24C3×C18C36C3×C12C3×C9C24C18C3×C6C12C9C32C6C3
# reps1162262418124182483672

Matrix representation of C3×C72 in GL2(𝔽73) generated by

80
08
,
240
020
G:=sub<GL(2,GF(73))| [8,0,0,8],[24,0,0,20] >;

C3×C72 in GAP, Magma, Sage, TeX

C_3\times C_{72}
% in TeX

G:=Group("C3xC72");
// GroupNames label

G:=SmallGroup(216,18);
// by ID

G=gap.SmallGroup(216,18);
# by ID

G:=PCGroup([6,-2,-3,-3,-2,-3,-2,108,223,165]);
// Polycyclic

G:=Group<a,b|a^3=b^72=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C3×C72 in TeX

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