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## G = C32×D20order 360 = 23·32·5

### Direct product of C32 and D20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C32×D20
 Chief series C1 — C5 — C10 — C30 — C3×C30 — D5×C3×C6 — C32×D20
 Lower central C5 — C10 — C32×D20
 Upper central C1 — C3×C6 — C3×C12

Generators and relations for C32×D20
G = < a,b,c,d | a3=b3=c20=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 288 in 96 conjugacy classes, 54 normal (14 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C6, D4, C32, D5, C10, C12, C2×C6, C15, C3×C6, C3×C6, C20, D10, C3×D4, C3×D5, C30, C3×C12, C62, D20, C3×C15, C60, C6×D5, D4×C32, C32×D5, C3×C30, C3×D20, C3×C60, D5×C3×C6, C32×D20
Quotients: C1, C2, C3, C22, C6, D4, C32, D5, C2×C6, C3×C6, D10, C3×D4, C3×D5, C62, D20, C6×D5, D4×C32, C32×D5, C3×D20, D5×C3×C6, C32×D20

Smallest permutation representation of C32×D20
On 180 points
Generators in S180
(1 34 60)(2 35 41)(3 36 42)(4 37 43)(5 38 44)(6 39 45)(7 40 46)(8 21 47)(9 22 48)(10 23 49)(11 24 50)(12 25 51)(13 26 52)(14 27 53)(15 28 54)(16 29 55)(17 30 56)(18 31 57)(19 32 58)(20 33 59)(61 85 157)(62 86 158)(63 87 159)(64 88 160)(65 89 141)(66 90 142)(67 91 143)(68 92 144)(69 93 145)(70 94 146)(71 95 147)(72 96 148)(73 97 149)(74 98 150)(75 99 151)(76 100 152)(77 81 153)(78 82 154)(79 83 155)(80 84 156)(101 163 139)(102 164 140)(103 165 121)(104 166 122)(105 167 123)(106 168 124)(107 169 125)(108 170 126)(109 171 127)(110 172 128)(111 173 129)(112 174 130)(113 175 131)(114 176 132)(115 177 133)(116 178 134)(117 179 135)(118 180 136)(119 161 137)(120 162 138)
(1 68 164)(2 69 165)(3 70 166)(4 71 167)(5 72 168)(6 73 169)(7 74 170)(8 75 171)(9 76 172)(10 77 173)(11 78 174)(12 79 175)(13 80 176)(14 61 177)(15 62 178)(16 63 179)(17 64 180)(18 65 161)(19 66 162)(20 67 163)(21 99 127)(22 100 128)(23 81 129)(24 82 130)(25 83 131)(26 84 132)(27 85 133)(28 86 134)(29 87 135)(30 88 136)(31 89 137)(32 90 138)(33 91 139)(34 92 140)(35 93 121)(36 94 122)(37 95 123)(38 96 124)(39 97 125)(40 98 126)(41 145 103)(42 146 104)(43 147 105)(44 148 106)(45 149 107)(46 150 108)(47 151 109)(48 152 110)(49 153 111)(50 154 112)(51 155 113)(52 156 114)(53 157 115)(54 158 116)(55 159 117)(56 160 118)(57 141 119)(58 142 120)(59 143 101)(60 144 102)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 26)(22 25)(23 24)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)(41 58)(42 57)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 50)(59 60)(61 74)(62 73)(63 72)(64 71)(65 70)(66 69)(67 68)(75 80)(76 79)(77 78)(81 82)(83 100)(84 99)(85 98)(86 97)(87 96)(88 95)(89 94)(90 93)(91 92)(101 102)(103 120)(104 119)(105 118)(106 117)(107 116)(108 115)(109 114)(110 113)(111 112)(121 138)(122 137)(123 136)(124 135)(125 134)(126 133)(127 132)(128 131)(129 130)(139 140)(141 146)(142 145)(143 144)(147 160)(148 159)(149 158)(150 157)(151 156)(152 155)(153 154)(161 166)(162 165)(163 164)(167 180)(168 179)(169 178)(170 177)(171 176)(172 175)(173 174)

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

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

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

117 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 4 5A 5B 6A ··· 6H 6I ··· 6X 10A 10B 12A ··· 12H 15A ··· 15P 20A 20B 20C 20D 30A ··· 30P 60A ··· 60AF order 1 2 2 2 3 ··· 3 4 5 5 6 ··· 6 6 ··· 6 10 10 12 ··· 12 15 ··· 15 20 20 20 20 30 ··· 30 60 ··· 60 size 1 1 10 10 1 ··· 1 2 2 2 1 ··· 1 10 ··· 10 2 2 2 ··· 2 2 ··· 2 2 2 2 2 2 ··· 2 2 ··· 2

117 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C3 C6 C6 D4 D5 D10 C3×D4 C3×D5 D20 C6×D5 C3×D20 kernel C32×D20 C3×C60 D5×C3×C6 C3×D20 C60 C6×D5 C3×C15 C3×C12 C3×C6 C15 C12 C32 C6 C3 # reps 1 1 2 8 8 16 1 2 2 8 16 4 16 32

Matrix representation of C32×D20 in GL3(𝔽61) generated by

 13 0 0 0 47 0 0 0 47
,
 1 0 0 0 13 0 0 0 13
,
 1 0 0 0 7 32 0 29 2
,
 60 0 0 0 7 32 0 29 54
G:=sub<GL(3,GF(61))| [13,0,0,0,47,0,0,0,47],[1,0,0,0,13,0,0,0,13],[1,0,0,0,7,29,0,32,2],[60,0,0,0,7,29,0,32,54] >;

C32×D20 in GAP, Magma, Sage, TeX

C_3^2\times D_{20}
% in TeX

G:=Group("C3^2xD20");
// GroupNames label

G:=SmallGroup(360,92);
// by ID

G=gap.SmallGroup(360,92);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-2,-5,457,223,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^20=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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