Copied to
clipboard

G = C3⋊D68order 408 = 23·3·17

The semidirect product of C3 and D68 acting via D68/D34=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C512D4, C32D68, D342S3, Dic3⋊D17, C6.5D34, C34.5D6, D1023C2, C102.5C22, (C6×D17)⋊2C2, C171(C3⋊D4), C2.5(S3×D17), (Dic3×C17)⋊3C2, SmallGroup(408,11)

Series: Derived Chief Lower central Upper central

C1C102 — C3⋊D68
C1C17C51C102C6×D17 — C3⋊D68
C51C102 — C3⋊D68
C1C2

Generators and relations for C3⋊D68
 G = < a,b,c | a3=b68=c2=1, bab-1=cac=a-1, cbc=b-1 >

34C2
102C2
3C4
17C22
51C22
34C6
34S3
2D17
6D17
51D4
17D6
17C2×C6
3C68
3D34
2C3×D17
2D51
17C3⋊D4
3D68

Smallest permutation representation of C3⋊D68
On 204 points
Generators in S204
(1 140 92)(2 93 141)(3 142 94)(4 95 143)(5 144 96)(6 97 145)(7 146 98)(8 99 147)(9 148 100)(10 101 149)(11 150 102)(12 103 151)(13 152 104)(14 105 153)(15 154 106)(16 107 155)(17 156 108)(18 109 157)(19 158 110)(20 111 159)(21 160 112)(22 113 161)(23 162 114)(24 115 163)(25 164 116)(26 117 165)(27 166 118)(28 119 167)(29 168 120)(30 121 169)(31 170 122)(32 123 171)(33 172 124)(34 125 173)(35 174 126)(36 127 175)(37 176 128)(38 129 177)(39 178 130)(40 131 179)(41 180 132)(42 133 181)(43 182 134)(44 135 183)(45 184 136)(46 69 185)(47 186 70)(48 71 187)(49 188 72)(50 73 189)(51 190 74)(52 75 191)(53 192 76)(54 77 193)(55 194 78)(56 79 195)(57 196 80)(58 81 197)(59 198 82)(60 83 199)(61 200 84)(62 85 201)(63 202 86)(64 87 203)(65 204 88)(66 89 137)(67 138 90)(68 91 139)
(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 51)(2 50)(3 49)(4 48)(5 47)(6 46)(7 45)(8 44)(9 43)(10 42)(11 41)(12 40)(13 39)(14 38)(15 37)(16 36)(17 35)(18 34)(19 33)(20 32)(21 31)(22 30)(23 29)(24 28)(25 27)(52 68)(53 67)(54 66)(55 65)(56 64)(57 63)(58 62)(59 61)(69 145)(70 144)(71 143)(72 142)(73 141)(74 140)(75 139)(76 138)(77 137)(78 204)(79 203)(80 202)(81 201)(82 200)(83 199)(84 198)(85 197)(86 196)(87 195)(88 194)(89 193)(90 192)(91 191)(92 190)(93 189)(94 188)(95 187)(96 186)(97 185)(98 184)(99 183)(100 182)(101 181)(102 180)(103 179)(104 178)(105 177)(106 176)(107 175)(108 174)(109 173)(110 172)(111 171)(112 170)(113 169)(114 168)(115 167)(116 166)(117 165)(118 164)(119 163)(120 162)(121 161)(122 160)(123 159)(124 158)(125 157)(126 156)(127 155)(128 154)(129 153)(130 152)(131 151)(132 150)(133 149)(134 148)(135 147)(136 146)

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

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

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

57 conjugacy classes

class 1 2A2B2C 3  4 6A6B6C17A···17H34A···34H51A···51H68A···68P102A···102H
order12223466617···1734···3451···5168···68102···102
size113410226234342···22···24···46···64···4

57 irreducible representations

dim1111222222244
type++++++++++++
imageC1C2C2C2S3D4D6C3⋊D4D17D34D68S3×D17C3⋊D68
kernelC3⋊D68Dic3×C17C6×D17D102D34C51C34C17Dic3C6C3C2C1
# reps11111112881688

Matrix representation of C3⋊D68 in GL4(𝔽409) generated by

0100
40840800
0010
0001
,
2376500
23717200
00192310
00135382
,
408000
1100
0057408
00385352
G:=sub<GL(4,GF(409))| [0,408,0,0,1,408,0,0,0,0,1,0,0,0,0,1],[237,237,0,0,65,172,0,0,0,0,192,135,0,0,310,382],[408,1,0,0,0,1,0,0,0,0,57,385,0,0,408,352] >;

C3⋊D68 in GAP, Magma, Sage, TeX

C_3\rtimes D_{68}
% in TeX

G:=Group("C3:D68");
// GroupNames label

G:=SmallGroup(408,11);
// by ID

G=gap.SmallGroup(408,11);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-17,61,26,168,9604]);
// Polycyclic

G:=Group<a,b,c|a^3=b^68=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3⋊D68 in TeX

׿
×
𝔽