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## G = C2×C92⋊4C3order 486 = 2·35

### Direct product of C2 and C92⋊4C3

direct product, metabelian, nilpotent (class 2), monomial, 3-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C92⋊4C3
 Chief series C1 — C3 — C32 — C3×C9 — C92 — C92⋊4C3 — C2×C92⋊4C3
 Lower central C1 — C32 — C2×C92⋊4C3
 Upper central C1 — C3×C6 — C2×C92⋊4C3

Generators and relations for C2×C924C3
G = < a,b,c,d | a2=b9=c9=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc6, dcd-1=b6c4 >

Subgroups: 180 in 100 conjugacy classes, 66 normal (10 characteristic)
C1, C2, C3, C3, C6, C6, C9, C32, C32, C18, C3×C6, C3×C6, C3×C9, C33, C3×C18, C32×C6, C92, C32⋊C9, C9⋊C9, C9×C18, C2×C32⋊C9, C2×C9⋊C9, C924C3, C2×C924C3
Quotients: C1, C2, C3, C6, C32, C3×C6, C33, C32×C6, C9○He3, C2×C9○He3, C924C3, C2×C924C3

Smallest permutation representation of C2×C924C3
On 162 points
Generators in S162
(1 100)(2 101)(3 102)(4 103)(5 104)(6 105)(7 106)(8 107)(9 108)(10 91)(11 92)(12 93)(13 94)(14 95)(15 96)(16 97)(17 98)(18 99)(19 139)(20 140)(21 141)(22 142)(23 143)(24 144)(25 136)(26 137)(27 138)(28 83)(29 84)(30 85)(31 86)(32 87)(33 88)(34 89)(35 90)(36 82)(37 118)(38 119)(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)(52 133)(53 134)(54 135)(55 110)(56 111)(57 112)(58 113)(59 114)(60 115)(61 116)(62 117)(63 109)(64 145)(65 146)(66 147)(67 148)(68 149)(69 150)(70 151)(71 152)(72 153)(73 154)(74 155)(75 156)(76 157)(77 158)(78 159)(79 160)(80 161)(81 162)
(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)
(1 22 44 10 113 88 77 70 53)(2 23 45 11 114 89 78 71 54)(3 24 37 12 115 90 79 72 46)(4 25 38 13 116 82 80 64 47)(5 26 39 14 117 83 81 65 48)(6 27 40 15 109 84 73 66 49)(7 19 41 16 110 85 74 67 50)(8 20 42 17 111 86 75 68 51)(9 21 43 18 112 87 76 69 52)(28 162 146 129 104 137 120 95 62)(29 154 147 130 105 138 121 96 63)(30 155 148 131 106 139 122 97 55)(31 156 149 132 107 140 123 98 56)(32 157 150 133 108 141 124 99 57)(33 158 151 134 100 142 125 91 58)(34 159 152 135 101 143 126 92 59)(35 160 153 127 102 144 118 93 60)(36 161 145 128 103 136 119 94 61)
(2 11 78)(3 79 12)(5 14 81)(6 73 15)(8 17 75)(9 76 18)(19 70 116)(20 23 26)(21 115 66)(22 64 110)(24 109 69)(25 67 113)(27 112 72)(28 126 132)(29 35 32)(30 128 125)(31 120 135)(33 131 119)(34 123 129)(36 134 122)(37 43 40)(38 88 50)(39 54 86)(41 82 53)(42 48 89)(44 85 47)(45 51 83)(46 52 49)(55 142 145)(56 59 62)(57 153 138)(58 136 148)(60 147 141)(61 139 151)(63 150 144)(65 68 71)(84 90 87)(92 159 101)(93 102 160)(95 162 104)(96 105 154)(98 156 107)(99 108 157)(111 114 117)(118 124 121)(127 133 130)(137 140 143)(146 149 152)

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

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

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

102 conjugacy classes

 class 1 2 3A ··· 3H 3I 3J 6A ··· 6H 6I 6J 9A ··· 9X 9Y ··· 9AN 18A ··· 18X 18Y ··· 18AN order 1 2 3 ··· 3 3 3 6 ··· 6 6 6 9 ··· 9 9 ··· 9 18 ··· 18 18 ··· 18 size 1 1 1 ··· 1 9 9 1 ··· 1 9 9 3 ··· 3 9 ··· 9 3 ··· 3 9 ··· 9

102 irreducible representations

 dim 1 1 1 1 1 1 1 1 3 3 type + + image C1 C2 C3 C3 C3 C6 C6 C6 C9○He3 C2×C9○He3 kernel C2×C92⋊4C3 C92⋊4C3 C9×C18 C2×C32⋊C9 C2×C9⋊C9 C92 C32⋊C9 C9⋊C9 C6 C3 # reps 1 1 2 8 16 2 8 16 24 24

Matrix representation of C2×C924C3 in GL6(𝔽19)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 18
,
 0 5 0 0 0 0 0 0 5 0 0 0 17 0 0 0 0 0 0 0 0 0 0 6 0 0 0 4 0 0 0 0 0 0 4 0
,
 0 1 0 0 0 0 0 0 1 0 0 0 11 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 7 0 0
,
 1 0 0 0 0 0 0 7 0 0 0 0 0 0 11 0 0 0 0 0 0 1 0 0 0 0 0 0 7 0 0 0 0 0 0 11

G:=sub<GL(6,GF(19))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[0,0,17,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,6,0,0],[0,0,11,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,7,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,11] >;

C2×C924C3 in GAP, Magma, Sage, TeX

C_2\times C_9^2\rtimes_4C_3
% in TeX

G:=Group("C2xC9^2:4C3");
// GroupNames label

G:=SmallGroup(486,203);
// by ID

G=gap.SmallGroup(486,203);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,548,1148,4113,93]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^9=c^9=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^6,d*c*d^-1=b^6*c^4>;
// generators/relations

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