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## G = C6×C36order 216 = 23·33

### Abelian group of type [6,36]

Aliases: C6×C36, SmallGroup(216,73)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6×C36
 Chief series C1 — C3 — C6 — C3×C6 — C3×C18 — C3×C36 — C6×C36
 Lower central C1 — C6×C36
 Upper central C1 — C6×C36

Generators and relations for C6×C36
G = < a,b | a6=b36=1, ab=ba >

Subgroups: 80, all normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C2×C4, C9, C32, C12, C2×C6, C2×C6, C18, C3×C6, C3×C6, C2×C12, C2×C12, C3×C9, C36, C2×C18, C3×C12, C62, C3×C18, C3×C18, C2×C36, C6×C12, C3×C36, C6×C18, C6×C36
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C9, C32, C12, C2×C6, C18, C3×C6, C2×C12, C3×C9, C36, C2×C18, C3×C12, C62, C3×C18, C2×C36, C6×C12, C3×C36, C6×C18, C6×C36

Smallest permutation representation of C6×C36
Regular action on 216 points
Generators in S216
(1 157 40 141 78 191)(2 158 41 142 79 192)(3 159 42 143 80 193)(4 160 43 144 81 194)(5 161 44 109 82 195)(6 162 45 110 83 196)(7 163 46 111 84 197)(8 164 47 112 85 198)(9 165 48 113 86 199)(10 166 49 114 87 200)(11 167 50 115 88 201)(12 168 51 116 89 202)(13 169 52 117 90 203)(14 170 53 118 91 204)(15 171 54 119 92 205)(16 172 55 120 93 206)(17 173 56 121 94 207)(18 174 57 122 95 208)(19 175 58 123 96 209)(20 176 59 124 97 210)(21 177 60 125 98 211)(22 178 61 126 99 212)(23 179 62 127 100 213)(24 180 63 128 101 214)(25 145 64 129 102 215)(26 146 65 130 103 216)(27 147 66 131 104 181)(28 148 67 132 105 182)(29 149 68 133 106 183)(30 150 69 134 107 184)(31 151 70 135 108 185)(32 152 71 136 73 186)(33 153 72 137 74 187)(34 154 37 138 75 188)(35 155 38 139 76 189)(36 156 39 140 77 190)
(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)| (1,157,40,141,78,191)(2,158,41,142,79,192)(3,159,42,143,80,193)(4,160,43,144,81,194)(5,161,44,109,82,195)(6,162,45,110,83,196)(7,163,46,111,84,197)(8,164,47,112,85,198)(9,165,48,113,86,199)(10,166,49,114,87,200)(11,167,50,115,88,201)(12,168,51,116,89,202)(13,169,52,117,90,203)(14,170,53,118,91,204)(15,171,54,119,92,205)(16,172,55,120,93,206)(17,173,56,121,94,207)(18,174,57,122,95,208)(19,175,58,123,96,209)(20,176,59,124,97,210)(21,177,60,125,98,211)(22,178,61,126,99,212)(23,179,62,127,100,213)(24,180,63,128,101,214)(25,145,64,129,102,215)(26,146,65,130,103,216)(27,147,66,131,104,181)(28,148,67,132,105,182)(29,149,68,133,106,183)(30,150,69,134,107,184)(31,151,70,135,108,185)(32,152,71,136,73,186)(33,153,72,137,74,187)(34,154,37,138,75,188)(35,155,38,139,76,189)(36,156,39,140,77,190), (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( (1,157,40,141,78,191)(2,158,41,142,79,192)(3,159,42,143,80,193)(4,160,43,144,81,194)(5,161,44,109,82,195)(6,162,45,110,83,196)(7,163,46,111,84,197)(8,164,47,112,85,198)(9,165,48,113,86,199)(10,166,49,114,87,200)(11,167,50,115,88,201)(12,168,51,116,89,202)(13,169,52,117,90,203)(14,170,53,118,91,204)(15,171,54,119,92,205)(16,172,55,120,93,206)(17,173,56,121,94,207)(18,174,57,122,95,208)(19,175,58,123,96,209)(20,176,59,124,97,210)(21,177,60,125,98,211)(22,178,61,126,99,212)(23,179,62,127,100,213)(24,180,63,128,101,214)(25,145,64,129,102,215)(26,146,65,130,103,216)(27,147,66,131,104,181)(28,148,67,132,105,182)(29,149,68,133,106,183)(30,150,69,134,107,184)(31,151,70,135,108,185)(32,152,71,136,73,186)(33,153,72,137,74,187)(34,154,37,138,75,188)(35,155,38,139,76,189)(36,156,39,140,77,190), (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([[(1,157,40,141,78,191),(2,158,41,142,79,192),(3,159,42,143,80,193),(4,160,43,144,81,194),(5,161,44,109,82,195),(6,162,45,110,83,196),(7,163,46,111,84,197),(8,164,47,112,85,198),(9,165,48,113,86,199),(10,166,49,114,87,200),(11,167,50,115,88,201),(12,168,51,116,89,202),(13,169,52,117,90,203),(14,170,53,118,91,204),(15,171,54,119,92,205),(16,172,55,120,93,206),(17,173,56,121,94,207),(18,174,57,122,95,208),(19,175,58,123,96,209),(20,176,59,124,97,210),(21,177,60,125,98,211),(22,178,61,126,99,212),(23,179,62,127,100,213),(24,180,63,128,101,214),(25,145,64,129,102,215),(26,146,65,130,103,216),(27,147,66,131,104,181),(28,148,67,132,105,182),(29,149,68,133,106,183),(30,150,69,134,107,184),(31,151,70,135,108,185),(32,152,71,136,73,186),(33,153,72,137,74,187),(34,154,37,138,75,188),(35,155,38,139,76,189),(36,156,39,140,77,190)], [(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)]])

C6×C36 is a maximal subgroup of   C36.69D6  C6.Dic18  C36⋊Dic3  C6.11D36  C36.70D6

216 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 4A 4B 4C 4D 6A ··· 6X 9A ··· 9R 12A ··· 12AF 18A ··· 18BB 36A ··· 36BT order 1 2 2 2 3 ··· 3 4 4 4 4 6 ··· 6 9 ··· 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 1 1 ··· 1 1 1 1 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1 1 ··· 1

216 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C3 C3 C4 C6 C6 C6 C6 C9 C12 C12 C18 C18 C36 kernel C6×C36 C3×C36 C6×C18 C2×C36 C6×C12 C3×C18 C36 C2×C18 C3×C12 C62 C2×C12 C18 C3×C6 C12 C2×C6 C6 # reps 1 2 1 6 2 4 12 6 4 2 18 24 8 36 18 72

Matrix representation of C6×C36 in GL3(𝔽37) generated by

 36 0 0 0 11 0 0 0 1
,
 1 0 0 0 29 0 0 0 3
G:=sub<GL(3,GF(37))| [36,0,0,0,11,0,0,0,1],[1,0,0,0,29,0,0,0,3] >;

C6×C36 in GAP, Magma, Sage, TeX

C_6\times C_{36}
% in TeX

G:=Group("C6xC36");
// GroupNames label

G:=SmallGroup(216,73);
// by ID

G=gap.SmallGroup(216,73);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-2,-3,216,338]);
// Polycyclic

G:=Group<a,b|a^6=b^36=1,a*b=b*a>;
// generators/relations

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