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G = C32×Dic5order 180 = 22·32·5

Direct product of C32 and Dic5

Aliases: C32×Dic5, C154C12, C30.4C6, (C3×C15)⋊9C4, C52(C3×C12), C10.(C3×C6), (C3×C6).3D5, C6.4(C3×D5), C2.(C32×D5), (C3×C30).3C2, SmallGroup(180,13)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C32×Dic5
 Chief series C1 — C5 — C10 — C30 — C3×C30 — C32×Dic5
 Lower central C5 — C32×Dic5
 Upper central C1 — C3×C6

Generators and relations for C32×Dic5
G = < a,b,c,d | a3=b3=c10=1, d2=c5, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Smallest permutation representation of C32×Dic5
Regular action on 180 points
Generators in S180
(1 89 49)(2 90 50)(3 81 41)(4 82 42)(5 83 43)(6 84 44)(7 85 45)(8 86 46)(9 87 47)(10 88 48)(11 61 51)(12 62 52)(13 63 53)(14 64 54)(15 65 55)(16 66 56)(17 67 57)(18 68 58)(19 69 59)(20 70 60)(21 71 31)(22 72 32)(23 73 33)(24 74 34)(25 75 35)(26 76 36)(27 77 37)(28 78 38)(29 79 39)(30 80 40)(91 171 131)(92 172 132)(93 173 133)(94 174 134)(95 175 135)(96 176 136)(97 177 137)(98 178 138)(99 179 139)(100 180 140)(101 151 141)(102 152 142)(103 153 143)(104 154 144)(105 155 145)(106 156 146)(107 157 147)(108 158 148)(109 159 149)(110 160 150)(111 161 121)(112 162 122)(113 163 123)(114 164 124)(115 165 125)(116 166 126)(117 167 127)(118 168 128)(119 169 129)(120 170 130)
(1 29 19)(2 30 20)(3 21 11)(4 22 12)(5 23 13)(6 24 14)(7 25 15)(8 26 16)(9 27 17)(10 28 18)(31 51 41)(32 52 42)(33 53 43)(34 54 44)(35 55 45)(36 56 46)(37 57 47)(38 58 48)(39 59 49)(40 60 50)(61 81 71)(62 82 72)(63 83 73)(64 84 74)(65 85 75)(66 86 76)(67 87 77)(68 88 78)(69 89 79)(70 90 80)(91 111 101)(92 112 102)(93 113 103)(94 114 104)(95 115 105)(96 116 106)(97 117 107)(98 118 108)(99 119 109)(100 120 110)(121 141 131)(122 142 132)(123 143 133)(124 144 134)(125 145 135)(126 146 136)(127 147 137)(128 148 138)(129 149 139)(130 150 140)(151 171 161)(152 172 162)(153 173 163)(154 174 164)(155 175 165)(156 176 166)(157 177 167)(158 178 168)(159 179 169)(160 180 170)
(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)(161 162 163 164 165 166 167 168 169 170)(171 172 173 174 175 176 177 178 179 180)
(1 96 6 91)(2 95 7 100)(3 94 8 99)(4 93 9 98)(5 92 10 97)(11 104 16 109)(12 103 17 108)(13 102 18 107)(14 101 19 106)(15 110 20 105)(21 114 26 119)(22 113 27 118)(23 112 28 117)(24 111 29 116)(25 120 30 115)(31 124 36 129)(32 123 37 128)(33 122 38 127)(34 121 39 126)(35 130 40 125)(41 134 46 139)(42 133 47 138)(43 132 48 137)(44 131 49 136)(45 140 50 135)(51 144 56 149)(52 143 57 148)(53 142 58 147)(54 141 59 146)(55 150 60 145)(61 154 66 159)(62 153 67 158)(63 152 68 157)(64 151 69 156)(65 160 70 155)(71 164 76 169)(72 163 77 168)(73 162 78 167)(74 161 79 166)(75 170 80 165)(81 174 86 179)(82 173 87 178)(83 172 88 177)(84 171 89 176)(85 180 90 175)

G:=sub<Sym(180)| (1,89,49)(2,90,50)(3,81,41)(4,82,42)(5,83,43)(6,84,44)(7,85,45)(8,86,46)(9,87,47)(10,88,48)(11,61,51)(12,62,52)(13,63,53)(14,64,54)(15,65,55)(16,66,56)(17,67,57)(18,68,58)(19,69,59)(20,70,60)(21,71,31)(22,72,32)(23,73,33)(24,74,34)(25,75,35)(26,76,36)(27,77,37)(28,78,38)(29,79,39)(30,80,40)(91,171,131)(92,172,132)(93,173,133)(94,174,134)(95,175,135)(96,176,136)(97,177,137)(98,178,138)(99,179,139)(100,180,140)(101,151,141)(102,152,142)(103,153,143)(104,154,144)(105,155,145)(106,156,146)(107,157,147)(108,158,148)(109,159,149)(110,160,150)(111,161,121)(112,162,122)(113,163,123)(114,164,124)(115,165,125)(116,166,126)(117,167,127)(118,168,128)(119,169,129)(120,170,130), (1,29,19)(2,30,20)(3,21,11)(4,22,12)(5,23,13)(6,24,14)(7,25,15)(8,26,16)(9,27,17)(10,28,18)(31,51,41)(32,52,42)(33,53,43)(34,54,44)(35,55,45)(36,56,46)(37,57,47)(38,58,48)(39,59,49)(40,60,50)(61,81,71)(62,82,72)(63,83,73)(64,84,74)(65,85,75)(66,86,76)(67,87,77)(68,88,78)(69,89,79)(70,90,80)(91,111,101)(92,112,102)(93,113,103)(94,114,104)(95,115,105)(96,116,106)(97,117,107)(98,118,108)(99,119,109)(100,120,110)(121,141,131)(122,142,132)(123,143,133)(124,144,134)(125,145,135)(126,146,136)(127,147,137)(128,148,138)(129,149,139)(130,150,140)(151,171,161)(152,172,162)(153,173,163)(154,174,164)(155,175,165)(156,176,166)(157,177,167)(158,178,168)(159,179,169)(160,180,170), (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)(161,162,163,164,165,166,167,168,169,170)(171,172,173,174,175,176,177,178,179,180), (1,96,6,91)(2,95,7,100)(3,94,8,99)(4,93,9,98)(5,92,10,97)(11,104,16,109)(12,103,17,108)(13,102,18,107)(14,101,19,106)(15,110,20,105)(21,114,26,119)(22,113,27,118)(23,112,28,117)(24,111,29,116)(25,120,30,115)(31,124,36,129)(32,123,37,128)(33,122,38,127)(34,121,39,126)(35,130,40,125)(41,134,46,139)(42,133,47,138)(43,132,48,137)(44,131,49,136)(45,140,50,135)(51,144,56,149)(52,143,57,148)(53,142,58,147)(54,141,59,146)(55,150,60,145)(61,154,66,159)(62,153,67,158)(63,152,68,157)(64,151,69,156)(65,160,70,155)(71,164,76,169)(72,163,77,168)(73,162,78,167)(74,161,79,166)(75,170,80,165)(81,174,86,179)(82,173,87,178)(83,172,88,177)(84,171,89,176)(85,180,90,175)>;

G:=Group( (1,89,49)(2,90,50)(3,81,41)(4,82,42)(5,83,43)(6,84,44)(7,85,45)(8,86,46)(9,87,47)(10,88,48)(11,61,51)(12,62,52)(13,63,53)(14,64,54)(15,65,55)(16,66,56)(17,67,57)(18,68,58)(19,69,59)(20,70,60)(21,71,31)(22,72,32)(23,73,33)(24,74,34)(25,75,35)(26,76,36)(27,77,37)(28,78,38)(29,79,39)(30,80,40)(91,171,131)(92,172,132)(93,173,133)(94,174,134)(95,175,135)(96,176,136)(97,177,137)(98,178,138)(99,179,139)(100,180,140)(101,151,141)(102,152,142)(103,153,143)(104,154,144)(105,155,145)(106,156,146)(107,157,147)(108,158,148)(109,159,149)(110,160,150)(111,161,121)(112,162,122)(113,163,123)(114,164,124)(115,165,125)(116,166,126)(117,167,127)(118,168,128)(119,169,129)(120,170,130), (1,29,19)(2,30,20)(3,21,11)(4,22,12)(5,23,13)(6,24,14)(7,25,15)(8,26,16)(9,27,17)(10,28,18)(31,51,41)(32,52,42)(33,53,43)(34,54,44)(35,55,45)(36,56,46)(37,57,47)(38,58,48)(39,59,49)(40,60,50)(61,81,71)(62,82,72)(63,83,73)(64,84,74)(65,85,75)(66,86,76)(67,87,77)(68,88,78)(69,89,79)(70,90,80)(91,111,101)(92,112,102)(93,113,103)(94,114,104)(95,115,105)(96,116,106)(97,117,107)(98,118,108)(99,119,109)(100,120,110)(121,141,131)(122,142,132)(123,143,133)(124,144,134)(125,145,135)(126,146,136)(127,147,137)(128,148,138)(129,149,139)(130,150,140)(151,171,161)(152,172,162)(153,173,163)(154,174,164)(155,175,165)(156,176,166)(157,177,167)(158,178,168)(159,179,169)(160,180,170), (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)(161,162,163,164,165,166,167,168,169,170)(171,172,173,174,175,176,177,178,179,180), (1,96,6,91)(2,95,7,100)(3,94,8,99)(4,93,9,98)(5,92,10,97)(11,104,16,109)(12,103,17,108)(13,102,18,107)(14,101,19,106)(15,110,20,105)(21,114,26,119)(22,113,27,118)(23,112,28,117)(24,111,29,116)(25,120,30,115)(31,124,36,129)(32,123,37,128)(33,122,38,127)(34,121,39,126)(35,130,40,125)(41,134,46,139)(42,133,47,138)(43,132,48,137)(44,131,49,136)(45,140,50,135)(51,144,56,149)(52,143,57,148)(53,142,58,147)(54,141,59,146)(55,150,60,145)(61,154,66,159)(62,153,67,158)(63,152,68,157)(64,151,69,156)(65,160,70,155)(71,164,76,169)(72,163,77,168)(73,162,78,167)(74,161,79,166)(75,170,80,165)(81,174,86,179)(82,173,87,178)(83,172,88,177)(84,171,89,176)(85,180,90,175) );

G=PermutationGroup([(1,89,49),(2,90,50),(3,81,41),(4,82,42),(5,83,43),(6,84,44),(7,85,45),(8,86,46),(9,87,47),(10,88,48),(11,61,51),(12,62,52),(13,63,53),(14,64,54),(15,65,55),(16,66,56),(17,67,57),(18,68,58),(19,69,59),(20,70,60),(21,71,31),(22,72,32),(23,73,33),(24,74,34),(25,75,35),(26,76,36),(27,77,37),(28,78,38),(29,79,39),(30,80,40),(91,171,131),(92,172,132),(93,173,133),(94,174,134),(95,175,135),(96,176,136),(97,177,137),(98,178,138),(99,179,139),(100,180,140),(101,151,141),(102,152,142),(103,153,143),(104,154,144),(105,155,145),(106,156,146),(107,157,147),(108,158,148),(109,159,149),(110,160,150),(111,161,121),(112,162,122),(113,163,123),(114,164,124),(115,165,125),(116,166,126),(117,167,127),(118,168,128),(119,169,129),(120,170,130)], [(1,29,19),(2,30,20),(3,21,11),(4,22,12),(5,23,13),(6,24,14),(7,25,15),(8,26,16),(9,27,17),(10,28,18),(31,51,41),(32,52,42),(33,53,43),(34,54,44),(35,55,45),(36,56,46),(37,57,47),(38,58,48),(39,59,49),(40,60,50),(61,81,71),(62,82,72),(63,83,73),(64,84,74),(65,85,75),(66,86,76),(67,87,77),(68,88,78),(69,89,79),(70,90,80),(91,111,101),(92,112,102),(93,113,103),(94,114,104),(95,115,105),(96,116,106),(97,117,107),(98,118,108),(99,119,109),(100,120,110),(121,141,131),(122,142,132),(123,143,133),(124,144,134),(125,145,135),(126,146,136),(127,147,137),(128,148,138),(129,149,139),(130,150,140),(151,171,161),(152,172,162),(153,173,163),(154,174,164),(155,175,165),(156,176,166),(157,177,167),(158,178,168),(159,179,169),(160,180,170)], [(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),(161,162,163,164,165,166,167,168,169,170),(171,172,173,174,175,176,177,178,179,180)], [(1,96,6,91),(2,95,7,100),(3,94,8,99),(4,93,9,98),(5,92,10,97),(11,104,16,109),(12,103,17,108),(13,102,18,107),(14,101,19,106),(15,110,20,105),(21,114,26,119),(22,113,27,118),(23,112,28,117),(24,111,29,116),(25,120,30,115),(31,124,36,129),(32,123,37,128),(33,122,38,127),(34,121,39,126),(35,130,40,125),(41,134,46,139),(42,133,47,138),(43,132,48,137),(44,131,49,136),(45,140,50,135),(51,144,56,149),(52,143,57,148),(53,142,58,147),(54,141,59,146),(55,150,60,145),(61,154,66,159),(62,153,67,158),(63,152,68,157),(64,151,69,156),(65,160,70,155),(71,164,76,169),(72,163,77,168),(73,162,78,167),(74,161,79,166),(75,170,80,165),(81,174,86,179),(82,173,87,178),(83,172,88,177),(84,171,89,176),(85,180,90,175)])

C32×Dic5 is a maximal subgroup of   C30.Dic3  C30.D6  C15⋊D12  C15⋊Dic6  D5×C3×C12

72 conjugacy classes

 class 1 2 3A ··· 3H 4A 4B 5A 5B 6A ··· 6H 10A 10B 12A ··· 12P 15A ··· 15P 30A ··· 30P order 1 2 3 ··· 3 4 4 5 5 6 ··· 6 10 10 12 ··· 12 15 ··· 15 30 ··· 30 size 1 1 1 ··· 1 5 5 2 2 1 ··· 1 2 2 5 ··· 5 2 ··· 2 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + - image C1 C2 C3 C4 C6 C12 D5 Dic5 C3×D5 C3×Dic5 kernel C32×Dic5 C3×C30 C3×Dic5 C3×C15 C30 C15 C3×C6 C32 C6 C3 # reps 1 1 8 2 8 16 2 2 16 16

Matrix representation of C32×Dic5 in GL3(𝔽61) generated by

 13 0 0 0 1 0 0 0 1
,
 13 0 0 0 47 0 0 0 47
,
 1 0 0 0 1 60 0 19 43
,
 1 0 0 0 50 11 0 0 11
G:=sub<GL(3,GF(61))| [13,0,0,0,1,0,0,0,1],[13,0,0,0,47,0,0,0,47],[1,0,0,0,1,19,0,60,43],[1,0,0,0,50,0,0,11,11] >;

C32×Dic5 in GAP, Magma, Sage, TeX

C_3^2\times {\rm Dic}_5
% in TeX

G:=Group("C3^2xDic5");
// GroupNames label

G:=SmallGroup(180,13);
// by ID

G=gap.SmallGroup(180,13);
# by ID

G:=PCGroup([5,-2,-3,-3,-2,-5,90,3604]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^10=1,d^2=c^5,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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