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## G = C2×C92⋊8C3order 486 = 2·35

### Direct product of C2 and C92⋊8C3

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C2×C92⋊8C3
 Chief series C1 — C3 — C32 — C3×C9 — C92 — C92⋊8C3 — C2×C92⋊8C3
 Lower central C1 — C32 — C2×C92⋊8C3
 Upper central C1 — C3×C6 — C2×C92⋊8C3

Generators and relations for C2×C928C3
G = < a,b,c,d | a2=b9=c9=d3=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc3, dcd-1=b6c7 >

Subgroups: 198 in 110 conjugacy classes, 72 normal (16 characteristic)
C1, C2, C3, C3, C3, C6, C6, C6, C9, C9, C32, C32, C18, C18, C3×C6, C3×C6, C3×C9, C3×C9, 3- 1+2, C33, C3×C18, C3×C18, C2×3- 1+2, C32×C6, C92, C32⋊C9, C9⋊C9, C3×3- 1+2, C9×C18, C2×C32⋊C9, C2×C9⋊C9, C6×3- 1+2, C928C3, C2×C928C3
Quotients: C1, C2, C3, C6, C32, C3×C6, 3- 1+2, C33, C2×3- 1+2, C32×C6, C3×3- 1+2, C9○He3, C6×3- 1+2, C2×C9○He3, C928C3, C2×C928C3

Smallest permutation representation of C2×C928C3
On 162 points
Generators in S162
(1 107)(2 108)(3 100)(4 101)(5 102)(6 103)(7 104)(8 105)(9 106)(10 144)(11 136)(12 137)(13 138)(14 139)(15 140)(16 141)(17 142)(18 143)(19 124)(20 125)(21 126)(22 118)(23 119)(24 120)(25 121)(26 122)(27 123)(28 83)(29 84)(30 85)(31 86)(32 87)(33 88)(34 89)(35 90)(36 82)(37 98)(38 99)(39 91)(40 92)(41 93)(42 94)(43 95)(44 96)(45 97)(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)
(1 113 86 77 17 20 37 70 53)(2 114 87 78 18 21 38 71 54)(3 115 88 79 10 22 39 72 46)(4 116 89 80 11 23 40 64 47)(5 117 90 81 12 24 41 65 48)(6 109 82 73 13 25 42 66 49)(7 110 83 74 14 26 43 67 50)(8 111 84 75 15 27 44 68 51)(9 112 85 76 16 19 45 69 52)(28 155 139 122 95 148 131 104 55)(29 156 140 123 96 149 132 105 56)(30 157 141 124 97 150 133 106 57)(31 158 142 125 98 151 134 107 58)(32 159 143 126 99 152 135 108 59)(33 160 144 118 91 153 127 100 60)(34 161 136 119 92 145 128 101 61)(35 162 137 120 93 146 129 102 62)(36 154 138 121 94 147 130 103 63)
(2 38 78)(3 79 39)(5 41 81)(6 73 42)(8 44 75)(9 76 45)(10 109 69)(11 67 113)(12 15 18)(13 112 72)(14 70 116)(16 115 66)(17 64 110)(19 25 22)(20 83 47)(21 51 90)(23 86 50)(24 54 84)(26 89 53)(27 48 87)(28 128 125)(29 120 135)(30 36 33)(31 131 119)(32 123 129)(34 134 122)(35 126 132)(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)(82 88 85)(91 100 160)(93 162 102)(94 103 154)(96 156 105)(97 106 157)(99 159 108)(111 114 117)(118 124 121)(127 133 130)(137 140 143)(146 149 152)

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

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

G=PermutationGroup([[(1,107),(2,108),(3,100),(4,101),(5,102),(6,103),(7,104),(8,105),(9,106),(10,144),(11,136),(12,137),(13,138),(14,139),(15,140),(16,141),(17,142),(18,143),(19,124),(20,125),(21,126),(22,118),(23,119),(24,120),(25,121),(26,122),(27,123),(28,83),(29,84),(30,85),(31,86),(32,87),(33,88),(34,89),(35,90),(36,82),(37,98),(38,99),(39,91),(40,92),(41,93),(42,94),(43,95),(44,96),(45,97),(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)], [(1,113,86,77,17,20,37,70,53),(2,114,87,78,18,21,38,71,54),(3,115,88,79,10,22,39,72,46),(4,116,89,80,11,23,40,64,47),(5,117,90,81,12,24,41,65,48),(6,109,82,73,13,25,42,66,49),(7,110,83,74,14,26,43,67,50),(8,111,84,75,15,27,44,68,51),(9,112,85,76,16,19,45,69,52),(28,155,139,122,95,148,131,104,55),(29,156,140,123,96,149,132,105,56),(30,157,141,124,97,150,133,106,57),(31,158,142,125,98,151,134,107,58),(32,159,143,126,99,152,135,108,59),(33,160,144,118,91,153,127,100,60),(34,161,136,119,92,145,128,101,61),(35,162,137,120,93,146,129,102,62),(36,154,138,121,94,147,130,103,63)], [(2,38,78),(3,79,39),(5,41,81),(6,73,42),(8,44,75),(9,76,45),(10,109,69),(11,67,113),(12,15,18),(13,112,72),(14,70,116),(16,115,66),(17,64,110),(19,25,22),(20,83,47),(21,51,90),(23,86,50),(24,54,84),(26,89,53),(27,48,87),(28,128,125),(29,120,135),(30,36,33),(31,131,119),(32,123,129),(34,134,122),(35,126,132),(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),(82,88,85),(91,100,160),(93,162,102),(94,103,154),(96,156,105),(97,106,157),(99,159,108),(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 1 1 3 3 3 3 type + + image C1 C2 C3 C3 C3 C3 C6 C6 C6 C6 3- 1+2 C2×3- 1+2 C9○He3 C2×C9○He3 kernel C2×C92⋊8C3 C92⋊8C3 C9×C18 C2×C32⋊C9 C2×C9⋊C9 C6×3- 1+2 C92 C32⋊C9 C9⋊C9 C3×3- 1+2 C18 C9 C6 C3 # reps 1 1 2 6 16 2 2 6 16 2 6 6 18 18

Matrix representation of C2×C928C3 in GL6(𝔽19)

 18 0 0 0 0 0 0 18 0 0 0 0 0 0 18 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 16 13 0 0 0 0 0 3 7 0 0 0 0 15 0 0 0 0 0 0 0 1 6 0 0 0 0 0 18 1 0 0 0 1 18 0
,
 5 10 0 0 0 0 0 14 1 0 0 0 0 13 0 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 5
,
 7 0 0 0 0 0 16 1 0 0 0 0 10 0 11 0 0 0 0 0 0 1 0 0 0 0 0 8 11 0 0 0 0 1 0 7

G:=sub<GL(6,GF(19))| [18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,13,3,15,0,0,0,0,7,0,0,0,0,0,0,0,1,0,1,0,0,0,6,18,18,0,0,0,0,1,0],[5,0,0,0,0,0,10,14,13,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[7,16,10,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,1,8,1,0,0,0,0,11,0,0,0,0,0,0,7] >;

C2×C928C3 in GAP, Magma, Sage, TeX

C_2\times C_9^2\rtimes_8C_3
% in TeX

G:=Group("C2xC9^2:8C3");
// GroupNames label

G:=SmallGroup(486,205);
// by ID

G=gap.SmallGroup(486,205);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-3,-3,548,1148,4113,165]);
// 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^3,d*c*d^-1=b^6*c^7>;
// generators/relations

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