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G = C36.17D6order 432 = 24·33

17th non-split extension by C36 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial

Aliases: C36.17D6, C12.17D18, D4.(C9⋊S3), (C3×D4).5D9, (D4×C9).5S3, (C3×C9)⋊12SD16, C93(D4.S3), C33(D4.D9), (C3×C18).47D4, (C3×C12).85D6, C12.D96C2, C36.S34C2, C6.24(C9⋊D4), (D4×C32).9S3, C18.24(C3⋊D4), (C3×C36).20C22, C3.(C329SD16), C6.16(C327D4), C32.5(D4.S3), C2.4(C6.D18), C4.1(C2×C9⋊S3), (D4×C3×C9).2C2, C12.1(C2×C3⋊S3), (C3×D4).1(C3⋊S3), (C3×C6).99(C3⋊D4), SmallGroup(432,190)

Series: Derived Chief Lower central Upper central

C1C3×C36 — C36.17D6
C1C3C32C3×C9C3×C18C3×C36C12.D9 — C36.17D6
C3×C9C3×C18C3×C36 — C36.17D6
C1C2C4D4

Generators and relations for C36.17D6
 G = < a,b,c | a36=b6=1, c2=a9, bab-1=a19, cac-1=a17, cbc-1=a27b-1 >

Subgroups: 480 in 100 conjugacy classes, 43 normal (19 characteristic)
C1, C2, C2, C3, C3 [×3], C4, C4, C22, C6, C6 [×3], C6 [×4], C8, D4, Q8, C9 [×3], C32, Dic3 [×4], C12, C12 [×3], C2×C6 [×4], SD16, C18 [×3], C18 [×3], C3×C6, C3×C6, C3⋊C8 [×4], Dic6 [×4], C3×D4, C3×D4 [×3], C3×C9, Dic9 [×3], C36 [×3], C2×C18 [×3], C3⋊Dic3, C3×C12, C62, D4.S3 [×4], C3×C18, C3×C18, C9⋊C8 [×3], Dic18 [×3], D4×C9 [×3], C324C8, C324Q8, D4×C32, C9⋊Dic3, C3×C36, C6×C18, D4.D9 [×3], C329SD16, C36.S3, C12.D9, D4×C3×C9, C36.17D6
Quotients: C1, C2 [×3], C22, S3 [×4], D4, D6 [×4], SD16, D9 [×3], C3⋊S3, C3⋊D4 [×4], D18 [×3], C2×C3⋊S3, D4.S3 [×4], C9⋊S3, C9⋊D4 [×3], C327D4, C2×C9⋊S3, D4.D9 [×3], C329SD16, C6.D18, C36.17D6

Smallest permutation representation of C36.17D6
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)
(1 153 68)(2 172 69 20 154 51)(3 155 70)(4 174 71 22 156 53)(5 157 72)(6 176 37 24 158 55)(7 159 38)(8 178 39 26 160 57)(9 161 40)(10 180 41 28 162 59)(11 163 42)(12 146 43 30 164 61)(13 165 44)(14 148 45 32 166 63)(15 167 46)(16 150 47 34 168 65)(17 169 48)(18 152 49 36 170 67)(19 171 50)(21 173 52)(23 175 54)(25 177 56)(27 179 58)(29 145 60)(31 147 62)(33 149 64)(35 151 66)(73 202 139 82 193 112)(74 185 140 101 194 131)(75 204 141 84 195 114)(76 187 142 103 196 133)(77 206 143 86 197 116)(78 189 144 105 198 135)(79 208 109 88 199 118)(80 191 110 107 200 137)(81 210 111 90 201 120)(83 212 113 92 203 122)(85 214 115 94 205 124)(87 216 117 96 207 126)(89 182 119 98 209 128)(91 184 121 100 211 130)(93 186 123 102 213 132)(95 188 125 104 215 134)(97 190 127 106 181 136)(99 192 129 108 183 138)
(1 215 10 188 19 197 28 206)(2 196 11 205 20 214 29 187)(3 213 12 186 21 195 30 204)(4 194 13 203 22 212 31 185)(5 211 14 184 23 193 32 202)(6 192 15 201 24 210 33 183)(7 209 16 182 25 191 34 200)(8 190 17 199 26 208 35 181)(9 207 18 216 27 189 36 198)(37 138 46 111 55 120 64 129)(38 119 47 128 56 137 65 110)(39 136 48 109 57 118 66 127)(40 117 49 126 58 135 67 144)(41 134 50 143 59 116 68 125)(42 115 51 124 60 133 69 142)(43 132 52 141 61 114 70 123)(44 113 53 122 62 131 71 140)(45 130 54 139 63 112 72 121)(73 148 82 157 91 166 100 175)(74 165 83 174 92 147 101 156)(75 146 84 155 93 164 102 173)(76 163 85 172 94 145 103 154)(77 180 86 153 95 162 104 171)(78 161 87 170 96 179 105 152)(79 178 88 151 97 160 106 169)(80 159 89 168 98 177 107 150)(81 176 90 149 99 158 108 167)

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

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

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

72 conjugacy classes

class 1 2A2B3A3B3C3D4A4B6A6B6C6D6E···6L8A8B9A···9I12A12B12C12D18A···18I18J···18AA36A···36I
order12233334466666···6889···91212121218···1818···1836···36
size1142222210822224···454542···244442···24···44···4

72 irreducible representations

dim111122222222222444
type+++++++++++---
imageC1C2C2C2S3S3D4D6D6SD16D9C3⋊D4C3⋊D4D18C9⋊D4D4.S3D4.S3D4.D9
kernelC36.17D6C36.S3C12.D9D4×C3×C9D4×C9D4×C32C3×C18C36C3×C12C3×C9C3×D4C18C3×C6C12C6C9C32C3
# reps1111311312962918319

Matrix representation of C36.17D6 in GL6(𝔽73)

69600000
0180000
0016000
0003200
0000142
00006672
,
64160000
0650000
001000
000100
000010
00006672
,
20150000
56530000
0006800
0029000
0000033
00003161

G:=sub<GL(6,GF(73))| [69,0,0,0,0,0,60,18,0,0,0,0,0,0,16,0,0,0,0,0,0,32,0,0,0,0,0,0,1,66,0,0,0,0,42,72],[64,0,0,0,0,0,16,65,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,66,0,0,0,0,0,72],[20,56,0,0,0,0,15,53,0,0,0,0,0,0,0,29,0,0,0,0,68,0,0,0,0,0,0,0,0,31,0,0,0,0,33,61] >;

C36.17D6 in GAP, Magma, Sage, TeX

C_{36}._{17}D_6
% in TeX

G:=Group("C36.17D6");
// GroupNames label

G:=SmallGroup(432,190);
// by ID

G=gap.SmallGroup(432,190);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,85,254,135,58,6164,662,4037,14118]);
// Polycyclic

G:=Group<a,b,c|a^36=b^6=1,c^2=a^9,b*a*b^-1=a^19,c*a*c^-1=a^17,c*b*c^-1=a^27*b^-1>;
// generators/relations

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