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G = C17×C3⋊S3order 306 = 2·32·17

Direct product of C17 and C3⋊S3

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C17×C3⋊S3, C513S3, C322C34, C3⋊(S3×C17), (C3×C51)⋊5C2, SmallGroup(306,8)

Series: Derived Chief Lower central Upper central

C1C32 — C17×C3⋊S3
C1C3C32C3×C51 — C17×C3⋊S3
C32 — C17×C3⋊S3
C1C17

Generators and relations for C17×C3⋊S3
 G = < a,b,c,d | a17=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

9C2
3S3
3S3
3S3
3S3
9C34
3S3×C17
3S3×C17
3S3×C17
3S3×C17

Smallest permutation representation of C17×C3⋊S3
On 153 points
Generators in S153
(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)
(1 88 74)(2 89 75)(3 90 76)(4 91 77)(5 92 78)(6 93 79)(7 94 80)(8 95 81)(9 96 82)(10 97 83)(11 98 84)(12 99 85)(13 100 69)(14 101 70)(15 102 71)(16 86 72)(17 87 73)(18 122 117)(19 123 118)(20 124 119)(21 125 103)(22 126 104)(23 127 105)(24 128 106)(25 129 107)(26 130 108)(27 131 109)(28 132 110)(29 133 111)(30 134 112)(31 135 113)(32 136 114)(33 120 115)(34 121 116)(35 52 142)(36 53 143)(37 54 144)(38 55 145)(39 56 146)(40 57 147)(41 58 148)(42 59 149)(43 60 150)(44 61 151)(45 62 152)(46 63 153)(47 64 137)(48 65 138)(49 66 139)(50 67 140)(51 68 141)
(1 147 18)(2 148 19)(3 149 20)(4 150 21)(5 151 22)(6 152 23)(7 153 24)(8 137 25)(9 138 26)(10 139 27)(11 140 28)(12 141 29)(13 142 30)(14 143 31)(15 144 32)(16 145 33)(17 146 34)(35 134 100)(36 135 101)(37 136 102)(38 120 86)(39 121 87)(40 122 88)(41 123 89)(42 124 90)(43 125 91)(44 126 92)(45 127 93)(46 128 94)(47 129 95)(48 130 96)(49 131 97)(50 132 98)(51 133 99)(52 112 69)(53 113 70)(54 114 71)(55 115 72)(56 116 73)(57 117 74)(58 118 75)(59 119 76)(60 103 77)(61 104 78)(62 105 79)(63 106 80)(64 107 81)(65 108 82)(66 109 83)(67 110 84)(68 111 85)
(18 147)(19 148)(20 149)(21 150)(22 151)(23 152)(24 153)(25 137)(26 138)(27 139)(28 140)(29 141)(30 142)(31 143)(32 144)(33 145)(34 146)(35 112)(36 113)(37 114)(38 115)(39 116)(40 117)(41 118)(42 119)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 134)(53 135)(54 136)(55 120)(56 121)(57 122)(58 123)(59 124)(60 125)(61 126)(62 127)(63 128)(64 129)(65 130)(66 131)(67 132)(68 133)(69 100)(70 101)(71 102)(72 86)(73 87)(74 88)(75 89)(76 90)(77 91)(78 92)(79 93)(80 94)(81 95)(82 96)(83 97)(84 98)(85 99)

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

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

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

102 conjugacy classes

class 1  2 3A3B3C3D17A···17P34A···34P51A···51BL
order12333317···1734···3451···51
size1922221···19···92···2

102 irreducible representations

dim111122
type+++
imageC1C2C17C34S3S3×C17
kernelC17×C3⋊S3C3×C51C3⋊S3C32C51C3
# reps111616464

Matrix representation of C17×C3⋊S3 in GL4(𝔽103) generated by

13000
01300
00130
00013
,
1000
0100
001021
001020
,
10210200
1000
000102
001102
,
1000
10210200
0001
0010
G:=sub<GL(4,GF(103))| [13,0,0,0,0,13,0,0,0,0,13,0,0,0,0,13],[1,0,0,0,0,1,0,0,0,0,102,102,0,0,1,0],[102,1,0,0,102,0,0,0,0,0,0,1,0,0,102,102],[1,102,0,0,0,102,0,0,0,0,0,1,0,0,1,0] >;

C17×C3⋊S3 in GAP, Magma, Sage, TeX

C_{17}\times C_3\rtimes S_3
% in TeX

G:=Group("C17xC3:S3");
// GroupNames label

G:=SmallGroup(306,8);
// by ID

G=gap.SmallGroup(306,8);
# by ID

G:=PCGroup([4,-2,-17,-3,-3,818,3267]);
// Polycyclic

G:=Group<a,b,c,d|a^17=b^3=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

Export

Subgroup lattice of C17×C3⋊S3 in TeX

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