Copied to
clipboard

G = C17xD12order 408 = 23·3·17

Direct product of C17 and D12

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

Aliases: C17xD12, C51:6D4, C68:3S3, C12:1C34, C204:5C2, D6:1C34, C34.15D6, C102.20C22, C4:(S3xC17), C3:1(D4xC17), (S3xC34):4C2, C2.4(S3xC34), C6.3(C2xC34), SmallGroup(408,22)

Series: Derived Chief Lower central Upper central

C1C6 — C17xD12
C1C3C6C102S3xC34 — C17xD12
C3C6 — C17xD12
C1C34C68

Generators and relations for C17xD12
 G = < a,b,c | a17=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 68 in 32 conjugacy classes, 18 normal (14 characteristic)
Quotients: C1, C2, C22, S3, D4, D6, C17, D12, C34, C2xC34, S3xC17, D4xC17, S3xC34, C17xD12
6C2
6C2
3C22
3C22
2S3
2S3
6C34
6C34
3D4
3C2xC34
3C2xC34
2S3xC17
2S3xC17
3D4xC17

Smallest permutation representation of C17xD12
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)
(1 201 32 51 181 80 64 169 145 89 113 132)(2 202 33 35 182 81 65 170 146 90 114 133)(3 203 34 36 183 82 66 154 147 91 115 134)(4 204 18 37 184 83 67 155 148 92 116 135)(5 188 19 38 185 84 68 156 149 93 117 136)(6 189 20 39 186 85 52 157 150 94 118 120)(7 190 21 40 187 69 53 158 151 95 119 121)(8 191 22 41 171 70 54 159 152 96 103 122)(9 192 23 42 172 71 55 160 153 97 104 123)(10 193 24 43 173 72 56 161 137 98 105 124)(11 194 25 44 174 73 57 162 138 99 106 125)(12 195 26 45 175 74 58 163 139 100 107 126)(13 196 27 46 176 75 59 164 140 101 108 127)(14 197 28 47 177 76 60 165 141 102 109 128)(15 198 29 48 178 77 61 166 142 86 110 129)(16 199 30 49 179 78 62 167 143 87 111 130)(17 200 31 50 180 79 63 168 144 88 112 131)
(1 32)(2 33)(3 34)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(17 31)(35 133)(36 134)(37 135)(38 136)(39 120)(40 121)(41 122)(42 123)(43 124)(44 125)(45 126)(46 127)(47 128)(48 129)(49 130)(50 131)(51 132)(52 150)(53 151)(54 152)(55 153)(56 137)(57 138)(58 139)(59 140)(60 141)(61 142)(62 143)(63 144)(64 145)(65 146)(66 147)(67 148)(68 149)(69 95)(70 96)(71 97)(72 98)(73 99)(74 100)(75 101)(76 102)(77 86)(78 87)(79 88)(80 89)(81 90)(82 91)(83 92)(84 93)(85 94)(103 171)(104 172)(105 173)(106 174)(107 175)(108 176)(109 177)(110 178)(111 179)(112 180)(113 181)(114 182)(115 183)(116 184)(117 185)(118 186)(119 187)

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

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

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

153 conjugacy classes

class 1 2A2B2C 3  4  6 12A12B17A···17P34A···34P34Q···34AV51A···51P68A···68P102A···102P204A···204AF
order1222346121217···1734···3434···3451···5168···68102···102204···204
size1166222221···11···16···62···22···22···22···2

153 irreducible representations

dim11111122222222
type+++++++
imageC1C2C2C17C34C34S3D4D6D12S3xC17D4xC17S3xC34C17xD12
kernelC17xD12C204S3xC34D12C12D6C68C51C34C17C4C3C2C1
# reps112161632111216161632

Matrix representation of C17xD12 in GL2(F409) generated by

360
036
,
35356
353297
,
11
0408
G:=sub<GL(2,GF(409))| [36,0,0,36],[353,353,56,297],[1,0,1,408] >;

C17xD12 in GAP, Magma, Sage, TeX

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

G:=Group("C17xD12");
// GroupNames label

G:=SmallGroup(408,22);
// by ID

G=gap.SmallGroup(408,22);
# by ID

G:=PCGroup([5,-2,-2,-17,-2,-3,701,346,6804]);
// Polycyclic

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

Export

Subgroup lattice of C17xD12 in TeX

׿
x
:
Z
F
o
wr
Q
<