Copied to
clipboard

G = C11×D17order 374 = 2·11·17

Direct product of C11 and D17

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

Aliases: C11×D17, C17⋊C22, C1872C2, SmallGroup(374,2)

Series: Derived Chief Lower central Upper central

C1C17 — C11×D17
C1C17C187 — C11×D17
C17 — C11×D17
C1C11

Generators and relations for C11×D17
 G = < a,b,c | a11=b17=c2=1, ab=ba, ac=ca, cbc=b-1 >

17C2
17C22

Smallest permutation representation of C11×D17
On 187 points
Generators in S187
(1 184 161 145 134 119 97 75 57 38 27)(2 185 162 146 135 103 98 76 58 39 28)(3 186 163 147 136 104 99 77 59 40 29)(4 187 164 148 120 105 100 78 60 41 30)(5 171 165 149 121 106 101 79 61 42 31)(6 172 166 150 122 107 102 80 62 43 32)(7 173 167 151 123 108 86 81 63 44 33)(8 174 168 152 124 109 87 82 64 45 34)(9 175 169 153 125 110 88 83 65 46 18)(10 176 170 137 126 111 89 84 66 47 19)(11 177 154 138 127 112 90 85 67 48 20)(12 178 155 139 128 113 91 69 68 49 21)(13 179 156 140 129 114 92 70 52 50 22)(14 180 157 141 130 115 93 71 53 51 23)(15 181 158 142 131 116 94 72 54 35 24)(16 182 159 143 132 117 95 73 55 36 25)(17 183 160 144 133 118 96 74 56 37 26)
(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)
(1 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)(35 40)(36 39)(37 38)(41 51)(42 50)(43 49)(44 48)(45 47)(52 61)(53 60)(54 59)(55 58)(56 57)(62 68)(63 67)(64 66)(69 80)(70 79)(71 78)(72 77)(73 76)(74 75)(81 85)(82 84)(86 90)(87 89)(91 102)(92 101)(93 100)(94 99)(95 98)(96 97)(103 117)(104 116)(105 115)(106 114)(107 113)(108 112)(109 111)(118 119)(120 130)(121 129)(122 128)(123 127)(124 126)(131 136)(132 135)(133 134)(137 152)(138 151)(139 150)(140 149)(141 148)(142 147)(143 146)(144 145)(154 167)(155 166)(156 165)(157 164)(158 163)(159 162)(160 161)(168 170)(171 179)(172 178)(173 177)(174 176)(180 187)(181 186)(182 185)(183 184)

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

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

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

110 conjugacy classes

class 1  2 11A···11J17A···17H22A···22J187A···187CB
order1211···1117···1722···22187···187
size1171···12···217···172···2

110 irreducible representations

dim111122
type+++
imageC1C2C11C22D17C11×D17
kernelC11×D17C187D17C17C11C1
# reps111010880

Matrix representation of C11×D17 in GL2(𝔽1123) generated by

1550
0155
,
2811
85344
,
985464
155138
G:=sub<GL(2,GF(1123))| [155,0,0,155],[281,85,1,344],[985,155,464,138] >;

C11×D17 in GAP, Magma, Sage, TeX

C_{11}\times D_{17}
% in TeX

G:=Group("C11xD17");
// GroupNames label

G:=SmallGroup(374,2);
// by ID

G=gap.SmallGroup(374,2);
# by ID

G:=PCGroup([3,-2,-11,-17,3170]);
// Polycyclic

G:=Group<a,b,c|a^11=b^17=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C11×D17 in TeX

׿
×
𝔽