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## G = C32×C3.A4order 324 = 22·34

### Direct product of C32 and C3.A4

Aliases: C32×C3.A4, C623C9, C33.3A4, C62.26C32, (C3×C62).3C3, C3.2(C32×A4), (C2×C6).8C33, C222(C32×C9), C32.22(C3×A4), (C2×C6)⋊2(C3×C9), SmallGroup(324,133)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C32×C3.A4
 Chief series C1 — C22 — C2×C6 — C3.A4 — C3×C3.A4 — C32×C3.A4
 Lower central C22 — C32×C3.A4
 Upper central C1 — C33

Generators and relations for C32×C3.A4
G = < a,b,c,d,e,f | a3=b3=c3=d2=e2=1, f3=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fdf-1=de=ed, fef-1=d >

Subgroups: 250 in 128 conjugacy classes, 78 normal (7 characteristic)
C1, C2, C3, C3, C22, C6, C9, C32, C2×C6, C2×C6, C3×C6, C3×C9, C33, C3.A4, C62, C32×C6, C32×C9, C3×C3.A4, C3×C62, C32×C3.A4
Quotients: C1, C3, C9, C32, A4, C3×C9, C33, C3.A4, C3×A4, C32×C9, C3×C3.A4, C32×A4, C32×C3.A4

Smallest permutation representation of C32×C3.A4
On 162 points
Generators in S162
(1 27 140)(2 19 141)(3 20 142)(4 21 143)(5 22 144)(6 23 136)(7 24 137)(8 25 138)(9 26 139)(10 56 130)(11 57 131)(12 58 132)(13 59 133)(14 60 134)(15 61 135)(16 62 127)(17 63 128)(18 55 129)(28 78 71)(29 79 72)(30 80 64)(31 81 65)(32 73 66)(33 74 67)(34 75 68)(35 76 69)(36 77 70)(37 103 149)(38 104 150)(39 105 151)(40 106 152)(41 107 153)(42 108 145)(43 100 146)(44 101 147)(45 102 148)(46 92 157)(47 93 158)(48 94 159)(49 95 160)(50 96 161)(51 97 162)(52 98 154)(53 99 155)(54 91 156)(82 124 114)(83 125 115)(84 126 116)(85 118 117)(86 119 109)(87 120 110)(88 121 111)(89 122 112)(90 123 113)
(1 82 100)(2 83 101)(3 84 102)(4 85 103)(5 86 104)(6 87 105)(7 88 106)(8 89 107)(9 90 108)(10 94 66)(11 95 67)(12 96 68)(13 97 69)(14 98 70)(15 99 71)(16 91 72)(17 92 64)(18 93 65)(19 125 147)(20 126 148)(21 118 149)(22 119 150)(23 120 151)(24 121 152)(25 122 153)(26 123 145)(27 124 146)(28 61 155)(29 62 156)(30 63 157)(31 55 158)(32 56 159)(33 57 160)(34 58 161)(35 59 162)(36 60 154)(37 143 117)(38 144 109)(39 136 110)(40 137 111)(41 138 112)(42 139 113)(43 140 114)(44 141 115)(45 142 116)(46 80 128)(47 81 129)(48 73 130)(49 74 131)(50 75 132)(51 76 133)(52 77 134)(53 78 135)(54 79 127)
(1 4 7)(2 5 8)(3 6 9)(10 13 16)(11 14 17)(12 15 18)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)(73 76 79)(74 77 80)(75 78 81)(82 85 88)(83 86 89)(84 87 90)(91 94 97)(92 95 98)(93 96 99)(100 103 106)(101 104 107)(102 105 108)(109 112 115)(110 113 116)(111 114 117)(118 121 124)(119 122 125)(120 123 126)(127 130 133)(128 131 134)(129 132 135)(136 139 142)(137 140 143)(138 141 144)(145 148 151)(146 149 152)(147 150 153)(154 157 160)(155 158 161)(156 159 162)
(2 161)(3 162)(5 155)(6 156)(8 158)(9 159)(10 42)(12 44)(13 45)(15 38)(16 39)(18 41)(19 50)(20 51)(22 53)(23 54)(25 47)(26 48)(28 86)(29 87)(31 89)(32 90)(34 83)(35 84)(55 107)(56 108)(58 101)(59 102)(61 104)(62 105)(65 112)(66 113)(68 115)(69 116)(71 109)(72 110)(73 123)(75 125)(76 126)(78 119)(79 120)(81 122)(91 136)(93 138)(94 139)(96 141)(97 142)(99 144)(127 151)(129 153)(130 145)(132 147)(133 148)(135 150)
(1 160)(3 162)(4 154)(6 156)(7 157)(9 159)(10 42)(11 43)(13 45)(14 37)(16 39)(17 40)(20 51)(21 52)(23 54)(24 46)(26 48)(27 49)(29 87)(30 88)(32 90)(33 82)(35 84)(36 85)(56 108)(57 100)(59 102)(60 103)(62 105)(63 106)(64 111)(66 113)(67 114)(69 116)(70 117)(72 110)(73 123)(74 124)(76 126)(77 118)(79 120)(80 121)(91 136)(92 137)(94 139)(95 140)(97 142)(98 143)(127 151)(128 152)(130 145)(131 146)(133 148)(134 149)
(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)

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

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

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

108 conjugacy classes

 class 1 2 3A ··· 3Z 6A ··· 6Z 9A ··· 9BB order 1 2 3 ··· 3 6 ··· 6 9 ··· 9 size 1 3 1 ··· 1 3 ··· 3 4 ··· 4

108 irreducible representations

 dim 1 1 1 1 3 3 3 type + + image C1 C3 C3 C9 A4 C3.A4 C3×A4 kernel C32×C3.A4 C3×C3.A4 C3×C62 C62 C33 C32 C32 # reps 1 24 2 54 1 18 8

Matrix representation of C32×C3.A4 in GL5(𝔽19)

 1 0 0 0 0 0 11 0 0 0 0 0 7 0 0 0 0 0 7 0 0 0 0 0 7
,
 11 0 0 0 0 0 1 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 7 0 0 0 0 0 7 0 0 0 0 0 7
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 18 0 0 0 2 0 18
,
 1 0 0 0 0 0 1 0 0 0 0 0 18 0 0 0 0 0 18 0 0 0 17 6 1
,
 11 0 0 0 0 0 7 0 0 0 0 0 0 11 0 0 0 3 10 16 0 0 0 0 9

G:=sub<GL(5,GF(19))| [1,0,0,0,0,0,11,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,7],[11,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,7],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,2,0,0,0,18,0,0,0,0,0,18],[1,0,0,0,0,0,1,0,0,0,0,0,18,0,17,0,0,0,18,6,0,0,0,0,1],[11,0,0,0,0,0,7,0,0,0,0,0,0,3,0,0,0,11,10,0,0,0,0,16,9] >;

C32×C3.A4 in GAP, Magma, Sage, TeX

C_3^2\times C_3.A_4
% in TeX

G:=Group("C3^2xC3.A4");
// GroupNames label

G:=SmallGroup(324,133);
// by ID

G=gap.SmallGroup(324,133);
# by ID

G:=PCGroup([6,-3,-3,-3,-3,-2,2,162,4864,8753]);
// Polycyclic

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

׿
×
𝔽