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G = C6×C36order 216 = 23·33

Abelian group of type [6,36]

direct product, abelian, monomial

Aliases: C6×C36, SmallGroup(216,73)

Series: Derived Chief Lower central Upper central

C1 — C6×C36
C1C3C6C3×C6C3×C18C3×C36 — C6×C36
C1 — C6×C36
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 [×2], C3, C3 [×3], C4 [×2], C22, C6, C6 [×11], C2×C4, C9 [×3], C32, C12 [×8], C2×C6, C2×C6 [×3], C18 [×9], C3×C6, C3×C6 [×2], C2×C12, C2×C12 [×3], C3×C9, C36 [×6], C2×C18 [×3], C3×C12 [×2], C62, C3×C18, C3×C18 [×2], C2×C36 [×3], C6×C12, C3×C36 [×2], C6×C18, C6×C36
Quotients: C1, C2 [×3], C3 [×4], C4 [×2], C22, C6 [×12], C2×C4, C9 [×3], C32, C12 [×8], C2×C6 [×4], C18 [×9], C3×C6 [×3], C2×C12 [×4], C3×C9, C36 [×6], C2×C18 [×3], C3×C12 [×2], C62, C3×C18 [×3], C2×C36 [×3], C6×C12, C3×C36 [×2], C6×C18, C6×C36

Smallest permutation representation of C6×C36
Regular action on 216 points
Generators in S216
(1 88 195 124 151 53)(2 89 196 125 152 54)(3 90 197 126 153 55)(4 91 198 127 154 56)(5 92 199 128 155 57)(6 93 200 129 156 58)(7 94 201 130 157 59)(8 95 202 131 158 60)(9 96 203 132 159 61)(10 97 204 133 160 62)(11 98 205 134 161 63)(12 99 206 135 162 64)(13 100 207 136 163 65)(14 101 208 137 164 66)(15 102 209 138 165 67)(16 103 210 139 166 68)(17 104 211 140 167 69)(18 105 212 141 168 70)(19 106 213 142 169 71)(20 107 214 143 170 72)(21 108 215 144 171 37)(22 73 216 109 172 38)(23 74 181 110 173 39)(24 75 182 111 174 40)(25 76 183 112 175 41)(26 77 184 113 176 42)(27 78 185 114 177 43)(28 79 186 115 178 44)(29 80 187 116 179 45)(30 81 188 117 180 46)(31 82 189 118 145 47)(32 83 190 119 146 48)(33 84 191 120 147 49)(34 85 192 121 148 50)(35 86 193 122 149 51)(36 87 194 123 150 52)
(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)| (1,88,195,124,151,53)(2,89,196,125,152,54)(3,90,197,126,153,55)(4,91,198,127,154,56)(5,92,199,128,155,57)(6,93,200,129,156,58)(7,94,201,130,157,59)(8,95,202,131,158,60)(9,96,203,132,159,61)(10,97,204,133,160,62)(11,98,205,134,161,63)(12,99,206,135,162,64)(13,100,207,136,163,65)(14,101,208,137,164,66)(15,102,209,138,165,67)(16,103,210,139,166,68)(17,104,211,140,167,69)(18,105,212,141,168,70)(19,106,213,142,169,71)(20,107,214,143,170,72)(21,108,215,144,171,37)(22,73,216,109,172,38)(23,74,181,110,173,39)(24,75,182,111,174,40)(25,76,183,112,175,41)(26,77,184,113,176,42)(27,78,185,114,177,43)(28,79,186,115,178,44)(29,80,187,116,179,45)(30,81,188,117,180,46)(31,82,189,118,145,47)(32,83,190,119,146,48)(33,84,191,120,147,49)(34,85,192,121,148,50)(35,86,193,122,149,51)(36,87,194,123,150,52), (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( (1,88,195,124,151,53)(2,89,196,125,152,54)(3,90,197,126,153,55)(4,91,198,127,154,56)(5,92,199,128,155,57)(6,93,200,129,156,58)(7,94,201,130,157,59)(8,95,202,131,158,60)(9,96,203,132,159,61)(10,97,204,133,160,62)(11,98,205,134,161,63)(12,99,206,135,162,64)(13,100,207,136,163,65)(14,101,208,137,164,66)(15,102,209,138,165,67)(16,103,210,139,166,68)(17,104,211,140,167,69)(18,105,212,141,168,70)(19,106,213,142,169,71)(20,107,214,143,170,72)(21,108,215,144,171,37)(22,73,216,109,172,38)(23,74,181,110,173,39)(24,75,182,111,174,40)(25,76,183,112,175,41)(26,77,184,113,176,42)(27,78,185,114,177,43)(28,79,186,115,178,44)(29,80,187,116,179,45)(30,81,188,117,180,46)(31,82,189,118,145,47)(32,83,190,119,146,48)(33,84,191,120,147,49)(34,85,192,121,148,50)(35,86,193,122,149,51)(36,87,194,123,150,52), (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([(1,88,195,124,151,53),(2,89,196,125,152,54),(3,90,197,126,153,55),(4,91,198,127,154,56),(5,92,199,128,155,57),(6,93,200,129,156,58),(7,94,201,130,157,59),(8,95,202,131,158,60),(9,96,203,132,159,61),(10,97,204,133,160,62),(11,98,205,134,161,63),(12,99,206,135,162,64),(13,100,207,136,163,65),(14,101,208,137,164,66),(15,102,209,138,165,67),(16,103,210,139,166,68),(17,104,211,140,167,69),(18,105,212,141,168,70),(19,106,213,142,169,71),(20,107,214,143,170,72),(21,108,215,144,171,37),(22,73,216,109,172,38),(23,74,181,110,173,39),(24,75,182,111,174,40),(25,76,183,112,175,41),(26,77,184,113,176,42),(27,78,185,114,177,43),(28,79,186,115,178,44),(29,80,187,116,179,45),(30,81,188,117,180,46),(31,82,189,118,145,47),(32,83,190,119,146,48),(33,84,191,120,147,49),(34,85,192,121,148,50),(35,86,193,122,149,51),(36,87,194,123,150,52)], [(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 2A2B2C3A···3H4A4B4C4D6A···6X9A···9R12A···12AF18A···18BB36A···36BT
order12223···344446···69···912···1218···1836···36
size11111···111111···11···11···11···11···1

216 irreducible representations

dim1111111111111111
type+++
imageC1C2C2C3C3C4C6C6C6C6C9C12C12C18C18C36
kernelC6×C36C3×C36C6×C18C2×C36C6×C12C3×C18C36C2×C18C3×C12C62C2×C12C18C3×C6C12C2×C6C6
# reps1216241264218248361872

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

3600
0110
001
,
100
0290
003
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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