S3xC2^2xC18 - GroupNames

G = S₃xC₂²xC₁₈ order 432 = 2⁴·3³

Direct product of C₂²xC₁₈ and S₃

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

Aliases: S₃xC₂²xC₁₈, C₆:(C₂²xC₁₈), C₃:(C₂³xC₁₈), (C₃xC₉):₂C₂⁴, (C₂²xC₆):₅C₁₈, (C₃xC₁₈):₂C₂³, (C₆xC₁₈):₁₀C₂², (C₂xC₆²).₂₈C₆, C₆².₆₉(C₂xC₆), C₃².₂(C₂³xC₆), (C₂xC₆xC₁₈):₃C₂, C₆.₇₂(S₃xC₂xC₆), (C₂xC₆):₆(C₂xC₁₈), (S₃xC₂xC₆).₁₀C₆, C₃.₄(S₃xC₂²xC₆), (C₂xC₆).₉₄(S₃xC₆), (C₃xS₃).(C₂²xC₆), (S₃xC₂²xC₆).₂C₃, (S₃xC₆).₂₁(C₂xC₆), (C₂²xC₆).₃₉(C₃xS₃), (C₃xC₆).₄₈(C₂²xC₆), SmallGroup(432,557)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃ — S₃xC₂²xC₁₈

Chief series

C₁ — C₃ — C₃² — C₃xC₉ — S₃xC₉ — S₃xC₁₈ — S₃xC₂xC₁₈ — S₃xC₂²xC₁₈

Lower central

C₃ — S₃xC₂²xC₁₈

Upper central

C₁ — C₂²xC₁₈

Generators and relations for S₃xC₂²xC₁₈
G = < a,b,c,d,e | a²=b²=c¹⁸=d³=e²=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d^-1 >

Subgroups: 772 in 434 conjugacy classes, 249 normal (15 characteristic)
C₁, C₂, C₂, C₃, C₃, C₂², C₂², S₃, C₆, C₆, C₂³, C₂³, C₉, C₉, C₃², D₆, C₂xC₆, C₂xC₆, C₂⁴, C₁₈, C₁₈, C₃xS₃, C₃xC₆, C₂²xS₃, C₂²xC₆, C₂²xC₆, C₃xC₉, C₂xC₁₈, C₂xC₁₈, S₃xC₆, C₆², S₃xC₂³, C₂³xC₆, S₃xC₉, C₃xC₁₈, C₂²xC₁₈, C₂²xC₁₈, S₃xC₂xC₆, C₂xC₆², S₃xC₁₈, C₆xC₁₈, C₂³xC₁₈, S₃xC₂²xC₆, S₃xC₂xC₁₈, C₂xC₆xC₁₈, S₃xC₂²xC₁₈
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, C₂³, C₉, D₆, C₂xC₆, C₂⁴, C₁₈, C₃xS₃, C₂²xS₃, C₂²xC₆, C₂xC₁₈, S₃xC₆, S₃xC₂³, C₂³xC₆, S₃xC₉, C₂²xC₁₈, S₃xC₂xC₆, S₃xC₁₈, C₂³xC₁₈, S₃xC₂²xC₆, S₃xC₂xC₁₈, S₃xC₂²xC₁₈

Smallest permutation representation of S₃xC₂²xC₁₈
►On 144 points

Generators in S₁₄₄

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

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

(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)

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

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

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

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

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

216 conjugacy classes

class	1	2A	···	2G	2H	···	2O	3A	3B	3C	3D	3E	6A	···	6N	6O	···	6AI	6AJ	···	6AY	9A	···	9F	9G	···	9L	18A	···	18AP	18AQ	···	18CF	18CG	···	18EB
order	1	2	···	2	2	···	2	3	3	3	3	3	6	···	6	6	···	6	6	···	6	9	···	9	9	···	9	18	···	18	18	···	18	18	···	18
size	1	1	···	1	3	···	3	1	1	2	2	2	1	···	1	2	···	2	3	···	3	1	···	1	2	···	2	1	···	1	2	···	2	3	···	3

216 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2
type	+	+	+							+	+
image	C₁	C₂	C₂	C₃	C₆	C₆	C₉	C₁₈	C₁₈	S₃	D₆	C₃xS₃	S₃xC₆	S₃xC₉	S₃xC₁₈
kernel	S₃xC₂²xC₁₈	S₃xC₂xC₁₈	C₂xC₆xC₁₈	S₃xC₂²xC₆	S₃xC₂xC₆	C₂xC₆²	S₃xC₂³	C₂²xS₃	C₂²xC₆	C₂²xC₁₈	C₂xC₁₈	C₂²xC₆	C₂xC₆	C₂³	C₂²
# reps	1	14	1	2	28	2	6	84	6	1	7	2	14	6	42

Matrix representation of S₃xC₂²xC₁₈ ►in GL₅(F₁₉)

18	0	0	0	0
0	18	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

1	0	0	0	0
0	1	0	0	0
0	0	18	0	0
0	0	0	1	0
0	0	0	0	1

4	0	0	0	0
0	13	0	0	0
0	0	6	0	0
0	0	0	6	0
0	0	0	0	6

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	11	0
0	0	0	0	7

18	0	0	0	0
0	18	0	0	0
0	0	1	0	0
0	0	0	0	1
0	0	0	1	0

G:=sub<GL(5,GF(19))| [18,0,0,0,0,0,18,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,18,0,0,0,0,0,1,0,0,0,0,0,1],[4,0,0,0,0,0,13,0,0,0,0,0,6,0,0,0,0,0,6,0,0,0,0,0,6],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,11,0,0,0,0,0,7],[18,0,0,0,0,0,18,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

S₃xC₂²xC₁₈ in GAP, Magma, Sage, TeX

S_3\times C_2^2\times C_{18}

% in TeX

G:=Group("S3xC2^2xC18");

// GroupNames label

G:=SmallGroup(432,557);

// by ID

G=gap.SmallGroup(432,557);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,137,14118]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^18=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;

// generators/relations

G = S3xC22xC18 order 432 = 24·33

Direct product of C22xC18 and S3

G = S₃xC₂²xC₁₈ order 432 = 2⁴·3³

Direct product of C₂²xC₁₈ and S₃