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## G = C4×C3⋊D15order 360 = 23·32·5

### Direct product of C4 and C3⋊D15

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C15 — C4×C3⋊D15
 Chief series C1 — C5 — C15 — C3×C15 — C3×C30 — C2×C3⋊D15 — C4×C3⋊D15
 Lower central C3×C15 — C4×C3⋊D15
 Upper central C1 — C4

Generators and relations for C4×C3⋊D15
G = < a,b,c,d | a4=b3=c15=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 696 in 96 conjugacy classes, 41 normal (15 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C2×C4, C32, D5, C10, Dic3, C12, D6, C15, C3⋊S3, C3×C6, Dic5, C20, D10, C4×S3, D15, C30, C3⋊Dic3, C3×C12, C2×C3⋊S3, C4×D5, C3×C15, Dic15, C60, D30, C4×C3⋊S3, C3⋊D15, C3×C30, C4×D15, C3⋊Dic15, C3×C60, C2×C3⋊D15, C4×C3⋊D15
Quotients: C1, C2, C4, C22, S3, C2×C4, D5, D6, C3⋊S3, D10, C4×S3, D15, C2×C3⋊S3, C4×D5, D30, C4×C3⋊S3, C3⋊D15, C4×D15, C2×C3⋊D15, C4×C3⋊D15

Smallest permutation representation of C4×C3⋊D15
On 180 points
Generators in S180
(1 143 55 91)(2 144 56 92)(3 145 57 93)(4 146 58 94)(5 147 59 95)(6 148 60 96)(7 149 46 97)(8 150 47 98)(9 136 48 99)(10 137 49 100)(11 138 50 101)(12 139 51 102)(13 140 52 103)(14 141 53 104)(15 142 54 105)(16 151 74 119)(17 152 75 120)(18 153 61 106)(19 154 62 107)(20 155 63 108)(21 156 64 109)(22 157 65 110)(23 158 66 111)(24 159 67 112)(25 160 68 113)(26 161 69 114)(27 162 70 115)(28 163 71 116)(29 164 72 117)(30 165 73 118)(31 168 78 123)(32 169 79 124)(33 170 80 125)(34 171 81 126)(35 172 82 127)(36 173 83 128)(37 174 84 129)(38 175 85 130)(39 176 86 131)(40 177 87 132)(41 178 88 133)(42 179 89 134)(43 180 90 135)(44 166 76 121)(45 167 77 122)
(1 39 30)(2 40 16)(3 41 17)(4 42 18)(5 43 19)(6 44 20)(7 45 21)(8 31 22)(9 32 23)(10 33 24)(11 34 25)(12 35 26)(13 36 27)(14 37 28)(15 38 29)(46 77 64)(47 78 65)(48 79 66)(49 80 67)(50 81 68)(51 82 69)(52 83 70)(53 84 71)(54 85 72)(55 86 73)(56 87 74)(57 88 75)(58 89 61)(59 90 62)(60 76 63)(91 131 118)(92 132 119)(93 133 120)(94 134 106)(95 135 107)(96 121 108)(97 122 109)(98 123 110)(99 124 111)(100 125 112)(101 126 113)(102 127 114)(103 128 115)(104 129 116)(105 130 117)(136 169 158)(137 170 159)(138 171 160)(139 172 161)(140 173 162)(141 174 163)(142 175 164)(143 176 165)(144 177 151)(145 178 152)(146 179 153)(147 180 154)(148 166 155)(149 167 156)(150 168 157)
(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 54)(2 53)(3 52)(4 51)(5 50)(6 49)(7 48)(8 47)(9 46)(10 60)(11 59)(12 58)(13 57)(14 56)(15 55)(16 84)(17 83)(18 82)(19 81)(20 80)(21 79)(22 78)(23 77)(24 76)(25 90)(26 89)(27 88)(28 87)(29 86)(30 85)(31 65)(32 64)(33 63)(34 62)(35 61)(36 75)(37 74)(38 73)(39 72)(40 71)(41 70)(42 69)(43 68)(44 67)(45 66)(91 142)(92 141)(93 140)(94 139)(95 138)(96 137)(97 136)(98 150)(99 149)(100 148)(101 147)(102 146)(103 145)(104 144)(105 143)(106 172)(107 171)(108 170)(109 169)(110 168)(111 167)(112 166)(113 180)(114 179)(115 178)(116 177)(117 176)(118 175)(119 174)(120 173)(121 159)(122 158)(123 157)(124 156)(125 155)(126 154)(127 153)(128 152)(129 151)(130 165)(131 164)(132 163)(133 162)(134 161)(135 160)

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

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

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

96 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 5A 5B 6A 6B 6C 6D 10A 10B 12A ··· 12H 15A ··· 15P 20A 20B 20C 20D 30A ··· 30P 60A ··· 60AF order 1 2 2 2 3 3 3 3 4 4 4 4 5 5 6 6 6 6 10 10 12 ··· 12 15 ··· 15 20 20 20 20 30 ··· 30 60 ··· 60 size 1 1 45 45 2 2 2 2 1 1 45 45 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 2 2 2 2 ··· 2 2 ··· 2

96 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C4 S3 D5 D6 D10 C4×S3 D15 C4×D5 D30 C4×D15 kernel C4×C3⋊D15 C3⋊Dic15 C3×C60 C2×C3⋊D15 C3⋊D15 C60 C3×C12 C30 C3×C6 C15 C12 C32 C6 C3 # reps 1 1 1 1 4 4 2 4 2 8 16 4 16 32

Matrix representation of C4×C3⋊D15 in GL4(𝔽61) generated by

 50 0 0 0 0 50 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 52 33 0 0 44 8
,
 9 5 0 0 56 38 0 0 0 0 0 44 0 0 18 43
,
 23 8 0 0 56 38 0 0 0 0 23 14 0 0 32 38
G:=sub<GL(4,GF(61))| [50,0,0,0,0,50,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,52,44,0,0,33,8],[9,56,0,0,5,38,0,0,0,0,0,18,0,0,44,43],[23,56,0,0,8,38,0,0,0,0,23,32,0,0,14,38] >;

C4×C3⋊D15 in GAP, Magma, Sage, TeX

C_4\times C_3\rtimes D_{15}
% in TeX

G:=Group("C4xC3:D15");
// GroupNames label

G:=SmallGroup(360,111);
// by ID

G=gap.SmallGroup(360,111);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-5,31,387,1444,10373]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^3=c^15=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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