Copied to
clipboard

G = C17⋊D12order 408 = 23·3·17

The semidirect product of C17 and D12 acting via D12/D6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C513D4, C172D12, D62D17, Dic17⋊S3, C6.6D34, C34.6D6, D1024C2, C102.6C22, (S3×C34)⋊2C2, C31(C17⋊D4), C2.6(S3×D17), (C3×Dic17)⋊3C2, SmallGroup(408,12)

Series: Derived Chief Lower central Upper central

C1C102 — C17⋊D12
C1C17C51C102C3×Dic17 — C17⋊D12
C51C102 — C17⋊D12
C1C2

Generators and relations for C17⋊D12
 G = < a,b,c | a17=b12=c2=1, bab-1=cac=a-1, cbc=b-1 >

6C2
102C2
3C22
17C4
51C22
2S3
34S3
6D17
6C34
51D4
17C12
17D6
3D34
3C2×C34
2S3×C17
2D51
17D12
3C17⋊D4

Smallest permutation representation of C17⋊D12
On 204 points
Generators in S204
(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)
(1 144 93 156 25 129 53 191 41 106 70 183)(2 143 94 155 26 128 54 190 42 105 71 182)(3 142 95 154 27 127 55 189 43 104 72 181)(4 141 96 170 28 126 56 188 44 103 73 180)(5 140 97 169 29 125 57 204 45 119 74 179)(6 139 98 168 30 124 58 203 46 118 75 178)(7 138 99 167 31 123 59 202 47 117 76 177)(8 137 100 166 32 122 60 201 48 116 77 176)(9 153 101 165 33 121 61 200 49 115 78 175)(10 152 102 164 34 120 62 199 50 114 79 174)(11 151 86 163 18 136 63 198 51 113 80 173)(12 150 87 162 19 135 64 197 35 112 81 172)(13 149 88 161 20 134 65 196 36 111 82 171)(14 148 89 160 21 133 66 195 37 110 83 187)(15 147 90 159 22 132 67 194 38 109 84 186)(16 146 91 158 23 131 68 193 39 108 85 185)(17 145 92 157 24 130 52 192 40 107 69 184)
(1 41)(2 40)(3 39)(4 38)(5 37)(6 36)(7 35)(8 51)(9 50)(10 49)(11 48)(12 47)(13 46)(14 45)(15 44)(16 43)(17 42)(18 32)(19 31)(20 30)(21 29)(22 28)(23 27)(24 26)(33 34)(52 94)(53 93)(54 92)(55 91)(56 90)(57 89)(58 88)(59 87)(60 86)(61 102)(62 101)(63 100)(64 99)(65 98)(66 97)(67 96)(68 95)(69 71)(72 85)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)(103 186)(104 185)(105 184)(106 183)(107 182)(108 181)(109 180)(110 179)(111 178)(112 177)(113 176)(114 175)(115 174)(116 173)(117 172)(118 171)(119 187)(120 165)(121 164)(122 163)(123 162)(124 161)(125 160)(126 159)(127 158)(128 157)(129 156)(130 155)(131 154)(132 170)(133 169)(134 168)(135 167)(136 166)(137 198)(138 197)(139 196)(140 195)(141 194)(142 193)(143 192)(144 191)(145 190)(146 189)(147 188)(148 204)(149 203)(150 202)(151 201)(152 200)(153 199)

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

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

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

57 conjugacy classes

class 1 2A2B2C 3  4  6 12A12B17A···17H34A···34H34I···34X51A···51H102A···102H
order1222346121217···1734···3434···3451···51102···102
size116102234234342···22···26···64···44···4

57 irreducible representations

dim1111222222244
type++++++++++++
imageC1C2C2C2S3D4D6D12D17D34C17⋊D4S3×D17C17⋊D12
kernelC17⋊D12C3×Dic17S3×C34D102Dic17C51C34C17D6C6C3C2C1
# reps11111112881688

Matrix representation of C17⋊D12 in GL4(𝔽409) generated by

83100
2022200
0010
0001
,
199500
33239000
00185
00178407
,
168200
40824100
00285
00178407
G:=sub<GL(4,GF(409))| [83,20,0,0,1,222,0,0,0,0,1,0,0,0,0,1],[19,332,0,0,95,390,0,0,0,0,1,178,0,0,85,407],[168,408,0,0,2,241,0,0,0,0,2,178,0,0,85,407] >;

C17⋊D12 in GAP, Magma, Sage, TeX

C_{17}\rtimes D_{12}
% in TeX

G:=Group("C17:D12");
// GroupNames label

G:=SmallGroup(408,12);
// by ID

G=gap.SmallGroup(408,12);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-17,20,61,168,9604]);
// Polycyclic

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

Export

Subgroup lattice of C17⋊D12 in TeX

׿
×
𝔽