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G = C36.S3order 216 = 23·33

6th non-split extension by C36 of S3 acting via S3/C3=C2

metabelian, supersoluble, monomial, A-group

Aliases: C36.6S3, C12.6D9, C6.3Dic9, C18.3Dic3, C3⋊(C9⋊C8), C9⋊(C3⋊C8), (C3×C9)⋊3C8, C4.2(C9⋊S3), (C3×C18).3C4, (C3×C36).5C2, C2.(C9⋊Dic3), C12.4(C3⋊S3), (C3×C12).16S3, C32.3(C3⋊C8), C3.(C324C8), (C3×C6).7Dic3, C6.1(C3⋊Dic3), SmallGroup(216,16)

Series: Derived Chief Lower central Upper central

C1C3×C9 — C36.S3
C1C3C32C3×C9C3×C18C3×C36 — C36.S3
C3×C9 — C36.S3
C1C4

Generators and relations for C36.S3
 G = < a,b,c | a36=1, b3=a12, c2=a9, ab=ba, cac-1=a17, cbc-1=a24b2 >

27C8
9C3⋊C8
9C3⋊C8
9C3⋊C8
9C3⋊C8
3C9⋊C8
3C324C8
3C9⋊C8
3C9⋊C8

Smallest permutation representation of C36.S3
Regular action on 216 points
Generators in S216
(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)
(1 208 124 13 184 136 25 196 112)(2 209 125 14 185 137 26 197 113)(3 210 126 15 186 138 27 198 114)(4 211 127 16 187 139 28 199 115)(5 212 128 17 188 140 29 200 116)(6 213 129 18 189 141 30 201 117)(7 214 130 19 190 142 31 202 118)(8 215 131 20 191 143 32 203 119)(9 216 132 21 192 144 33 204 120)(10 181 133 22 193 109 34 205 121)(11 182 134 23 194 110 35 206 122)(12 183 135 24 195 111 36 207 123)(37 108 161 49 84 173 61 96 149)(38 73 162 50 85 174 62 97 150)(39 74 163 51 86 175 63 98 151)(40 75 164 52 87 176 64 99 152)(41 76 165 53 88 177 65 100 153)(42 77 166 54 89 178 66 101 154)(43 78 167 55 90 179 67 102 155)(44 79 168 56 91 180 68 103 156)(45 80 169 57 92 145 69 104 157)(46 81 170 58 93 146 70 105 158)(47 82 171 59 94 147 71 106 159)(48 83 172 60 95 148 72 107 160)
(1 52 10 61 19 70 28 43)(2 69 11 42 20 51 29 60)(3 50 12 59 21 68 30 41)(4 67 13 40 22 49 31 58)(5 48 14 57 23 66 32 39)(6 65 15 38 24 47 33 56)(7 46 16 55 25 64 34 37)(8 63 17 72 26 45 35 54)(9 44 18 53 27 62 36 71)(73 135 82 144 91 117 100 126)(74 116 83 125 92 134 101 143)(75 133 84 142 93 115 102 124)(76 114 85 123 94 132 103 141)(77 131 86 140 95 113 104 122)(78 112 87 121 96 130 105 139)(79 129 88 138 97 111 106 120)(80 110 89 119 98 128 107 137)(81 127 90 136 99 109 108 118)(145 182 154 191 163 200 172 209)(146 199 155 208 164 181 173 190)(147 216 156 189 165 198 174 207)(148 197 157 206 166 215 175 188)(149 214 158 187 167 196 176 205)(150 195 159 204 168 213 177 186)(151 212 160 185 169 194 178 203)(152 193 161 202 170 211 179 184)(153 210 162 183 171 192 180 201)

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

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

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

C36.S3 is a maximal subgroup of
D9×C3⋊C8  C36.39D6  S3×C9⋊C8  D6.Dic9  D36⋊S3  D12.D9  Dic6⋊D9  C12.D18  C8×C9⋊S3  C72⋊S3  C36.69D6  C36.17D6  C36.18D6  C36.19D6  C36.20D6
C36.S3 is a maximal quotient of
C72.S3

60 conjugacy classes

class 1  2 3A3B3C3D4A4B6A6B6C6D8A8B8C8D9A···9I12A···12H18A···18I36A···36R
order12333344666688889···912···1218···1836···36
size112222112222272727272···22···22···22···2

60 irreducible representations

dim1111222222222
type++++--+-
imageC1C2C4C8S3S3Dic3Dic3D9C3⋊C8C3⋊C8Dic9C9⋊C8
kernelC36.S3C3×C36C3×C18C3×C9C36C3×C12C18C3×C6C12C9C32C6C3
# reps11243131962918

Matrix representation of C36.S3 in GL4(𝔽73) generated by

1000
0100
003465
00826
,
727200
1000
004531
00423
,
26200
607100
004229
007131
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,34,8,0,0,65,26],[72,1,0,0,72,0,0,0,0,0,45,42,0,0,31,3],[2,60,0,0,62,71,0,0,0,0,42,71,0,0,29,31] >;

C36.S3 in GAP, Magma, Sage, TeX

C_{36}.S_3
% in TeX

G:=Group("C36.S3");
// GroupNames label

G:=SmallGroup(216,16);
// by ID

G=gap.SmallGroup(216,16);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,12,31,2115,453,1444,5189]);
// Polycyclic

G:=Group<a,b,c|a^36=1,b^3=a^12,c^2=a^9,a*b=b*a,c*a*c^-1=a^17,c*b*c^-1=a^24*b^2>;
// generators/relations

Export

Subgroup lattice of C36.S3 in TeX

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