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## G = C3⋊S3×C20order 360 = 23·32·5

### Direct product of C20 and C3⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊S3×C20
 Chief series C1 — C3 — C32 — C3×C6 — C3×C30 — C10×C3⋊S3 — C3⋊S3×C20
 Lower central C32 — C3⋊S3×C20
 Upper central C1 — C20

Generators and relations for C3⋊S3×C20
G = < a,b,c,d | a20=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 256 in 96 conjugacy classes, 46 normal (18 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C2×C4, C32, C10, C10, Dic3, C12, D6, C15, C3⋊S3, C3×C6, C20, C20, C2×C10, C4×S3, C5×S3, C30, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C20, C3×C15, C5×Dic3, C60, S3×C10, C4×C3⋊S3, C5×C3⋊S3, C3×C30, S3×C20, C5×C3⋊Dic3, C3×C60, C10×C3⋊S3, C3⋊S3×C20
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, C10, D6, C3⋊S3, C20, C2×C10, C4×S3, C5×S3, C2×C3⋊S3, C2×C20, S3×C10, C4×C3⋊S3, C5×C3⋊S3, S3×C20, C10×C3⋊S3, C3⋊S3×C20

Smallest permutation representation of C3⋊S3×C20
On 180 points
Generators in S180
(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 71 164)(2 72 165)(3 73 166)(4 74 167)(5 75 168)(6 76 169)(7 77 170)(8 78 171)(9 79 172)(10 80 173)(11 61 174)(12 62 175)(13 63 176)(14 64 177)(15 65 178)(16 66 179)(17 67 180)(18 68 161)(19 69 162)(20 70 163)(21 155 111)(22 156 112)(23 157 113)(24 158 114)(25 159 115)(26 160 116)(27 141 117)(28 142 118)(29 143 119)(30 144 120)(31 145 101)(32 146 102)(33 147 103)(34 148 104)(35 149 105)(36 150 106)(37 151 107)(38 152 108)(39 153 109)(40 154 110)(41 128 88)(42 129 89)(43 130 90)(44 131 91)(45 132 92)(46 133 93)(47 134 94)(48 135 95)(49 136 96)(50 137 97)(51 138 98)(52 139 99)(53 140 100)(54 121 81)(55 122 82)(56 123 83)(57 124 84)(58 125 85)(59 126 86)(60 127 87)
(1 32 99)(2 33 100)(3 34 81)(4 35 82)(5 36 83)(6 37 84)(7 38 85)(8 39 86)(9 40 87)(10 21 88)(11 22 89)(12 23 90)(13 24 91)(14 25 92)(15 26 93)(16 27 94)(17 28 95)(18 29 96)(19 30 97)(20 31 98)(41 80 155)(42 61 156)(43 62 157)(44 63 158)(45 64 159)(46 65 160)(47 66 141)(48 67 142)(49 68 143)(50 69 144)(51 70 145)(52 71 146)(53 72 147)(54 73 148)(55 74 149)(56 75 150)(57 76 151)(58 77 152)(59 78 153)(60 79 154)(101 138 163)(102 139 164)(103 140 165)(104 121 166)(105 122 167)(106 123 168)(107 124 169)(108 125 170)(109 126 171)(110 127 172)(111 128 173)(112 129 174)(113 130 175)(114 131 176)(115 132 177)(116 133 178)(117 134 179)(118 135 180)(119 136 161)(120 137 162)
(21 88)(22 89)(23 90)(24 91)(25 92)(26 93)(27 94)(28 95)(29 96)(30 97)(31 98)(32 99)(33 100)(34 81)(35 82)(36 83)(37 84)(38 85)(39 86)(40 87)(41 111)(42 112)(43 113)(44 114)(45 115)(46 116)(47 117)(48 118)(49 119)(50 120)(51 101)(52 102)(53 103)(54 104)(55 105)(56 106)(57 107)(58 108)(59 109)(60 110)(61 174)(62 175)(63 176)(64 177)(65 178)(66 179)(67 180)(68 161)(69 162)(70 163)(71 164)(72 165)(73 166)(74 167)(75 168)(76 169)(77 170)(78 171)(79 172)(80 173)(121 148)(122 149)(123 150)(124 151)(125 152)(126 153)(127 154)(128 155)(129 156)(130 157)(131 158)(132 159)(133 160)(134 141)(135 142)(136 143)(137 144)(138 145)(139 146)(140 147)

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

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

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

120 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 5A 5B 5C 5D 6A 6B 6C 6D 10A 10B 10C 10D 10E ··· 10L 12A ··· 12H 15A ··· 15P 20A ··· 20H 20I ··· 20P 30A ··· 30P 60A ··· 60AF order 1 2 2 2 3 3 3 3 4 4 4 4 5 5 5 5 6 6 6 6 10 10 10 10 10 ··· 10 12 ··· 12 15 ··· 15 20 ··· 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 9 9 2 2 2 2 1 1 9 9 1 1 1 1 2 2 2 2 1 1 1 1 9 ··· 9 2 ··· 2 2 ··· 2 1 ··· 1 9 ··· 9 2 ··· 2 2 ··· 2

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 C5 C10 C10 C10 C20 S3 D6 C4×S3 C5×S3 S3×C10 S3×C20 kernel C3⋊S3×C20 C5×C3⋊Dic3 C3×C60 C10×C3⋊S3 C5×C3⋊S3 C4×C3⋊S3 C3⋊Dic3 C3×C12 C2×C3⋊S3 C3⋊S3 C60 C30 C15 C12 C6 C3 # reps 1 1 1 1 4 4 4 4 4 16 4 4 8 16 16 32

Matrix representation of C3⋊S3×C20 in GL4(𝔽61) generated by

 38 0 0 0 0 38 0 0 0 0 60 0 0 0 0 60
,
 0 1 0 0 60 60 0 0 0 0 0 1 0 0 60 60
,
 0 1 0 0 60 60 0 0 0 0 60 60 0 0 1 0
,
 1 0 0 0 60 60 0 0 0 0 60 0 0 0 1 1
G:=sub<GL(4,GF(61))| [38,0,0,0,0,38,0,0,0,0,60,0,0,0,0,60],[0,60,0,0,1,60,0,0,0,0,0,60,0,0,1,60],[0,60,0,0,1,60,0,0,0,0,60,1,0,0,60,0],[1,60,0,0,0,60,0,0,0,0,60,1,0,0,0,1] >;

C3⋊S3×C20 in GAP, Magma, Sage, TeX

C_3\rtimes S_3\times C_{20}
% in TeX

G:=Group("C3:S3xC20");
// GroupNames label

G:=SmallGroup(360,106);
// by ID

G=gap.SmallGroup(360,106);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-3,-3,127,2404,8645]);
// Polycyclic

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

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