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G = C3×D68order 408 = 23·3·17

Direct product of C3 and D68

direct product, metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C3×D68, C515D4, C681C6, C2043C2, D341C6, C123D17, C6.15D34, C102.15C22, C4⋊(C3×D17), C171(C3×D4), (C6×D17)⋊4C2, C34.3(C2×C6), C2.4(C6×D17), SmallGroup(408,17)

Series: Derived Chief Lower central Upper central

C1C34 — C3×D68
C1C17C34C102C6×D17 — C3×D68
C17C34 — C3×D68
C1C6C12

Generators and relations for C3×D68
 G = < a,b,c | a3=b68=c2=1, ab=ba, ac=ca, cbc=b-1 >

34C2
34C2
17C22
17C22
34C6
34C6
2D17
2D17
17D4
17C2×C6
17C2×C6
2C3×D17
2C3×D17
17C3×D4

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

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

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

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

111 conjugacy classes

class 1 2A2B2C3A3B 4 6A6B6C6D6E6F12A12B17A···17H34A···34H51A···51P68A···68P102A···102P204A···204AF
order1222334666666121217···1734···3451···5168···68102···102204···204
size1134341121134343434222···22···22···22···22···22···2

111 irreducible representations

dim11111122222222
type+++++++
imageC1C2C2C3C6C6D4C3×D4D17D34C3×D17D68C6×D17C3×D68
kernelC3×D68C204C6×D17D68C68D34C51C17C12C6C4C3C2C1
# reps112224128816161632

Matrix representation of C3×D68 in GL2(𝔽409) generated by

530
053
,
294230
17983
,
181305
315228
G:=sub<GL(2,GF(409))| [53,0,0,53],[294,179,230,83],[181,315,305,228] >;

C3×D68 in GAP, Magma, Sage, TeX

C_3\times D_{68}
% in TeX

G:=Group("C3xD68");
// GroupNames label

G:=SmallGroup(408,17);
// by ID

G=gap.SmallGroup(408,17);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-17,141,66,9604]);
// Polycyclic

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

Export

Subgroup lattice of C3×D68 in TeX

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