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## G = D5×C62order 360 = 23·32·5

### Direct product of C62 and D5

Aliases: D5×C62, C10⋊C62, C5⋊(C2×C62), C303(C2×C6), (C6×C30)⋊7C2, (C2×C30)⋊9C6, (C3×C15)⋊9C23, C153(C22×C6), (C3×C30)⋊8C22, (C2×C10)⋊5(C3×C6), SmallGroup(360,157)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — D5×C62
 Chief series C1 — C5 — C15 — C3×C15 — C32×D5 — D5×C3×C6 — D5×C62
 Lower central C5 — D5×C62
 Upper central C1 — C62

Generators and relations for D5×C62
G = < a,b,c,d | a6=b6=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 456 in 192 conjugacy classes, 126 normal (10 characteristic)
C1, C2, C2, C3, C22, C22, C5, C6, C6, C23, C32, D5, C10, C2×C6, C2×C6, C15, C3×C6, C3×C6, D10, C2×C10, C22×C6, C3×D5, C30, C62, C62, C22×D5, C3×C15, C6×D5, C2×C30, C2×C62, C32×D5, C3×C30, D5×C2×C6, D5×C3×C6, C6×C30, D5×C62
Quotients: C1, C2, C3, C22, C6, C23, C32, D5, C2×C6, C3×C6, D10, C22×C6, C3×D5, C62, C22×D5, C6×D5, C2×C62, C32×D5, D5×C2×C6, D5×C3×C6, D5×C62

Smallest permutation representation of D5×C62
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 36 99 89 50 169)(2 31 100 90 51 170)(3 32 101 85 52 171)(4 33 102 86 53 172)(5 34 97 87 54 173)(6 35 98 88 49 174)(7 106 96 73 48 22)(8 107 91 74 43 23)(9 108 92 75 44 24)(10 103 93 76 45 19)(11 104 94 77 46 20)(12 105 95 78 47 21)(13 28 130 83 126 151)(14 29 131 84 121 152)(15 30 132 79 122 153)(16 25 127 80 123 154)(17 26 128 81 124 155)(18 27 129 82 125 156)(37 168 144 111 118 62)(38 163 139 112 119 63)(39 164 140 113 120 64)(40 165 141 114 115 65)(41 166 142 109 116 66)(42 167 143 110 117 61)(55 176 147 162 70 136)(56 177 148 157 71 137)(57 178 149 158 72 138)(58 179 150 159 67 133)(59 180 145 160 68 134)(60 175 146 161 69 135)
(1 151 137 143 104)(2 152 138 144 105)(3 153 133 139 106)(4 154 134 140 107)(5 155 135 141 108)(6 156 136 142 103)(7 171 122 67 163)(8 172 123 68 164)(9 173 124 69 165)(10 174 125 70 166)(11 169 126 71 167)(12 170 121 72 168)(13 56 110 94 36)(14 57 111 95 31)(15 58 112 96 32)(16 59 113 91 33)(17 60 114 92 34)(18 55 109 93 35)(19 49 82 162 41)(20 50 83 157 42)(21 51 84 158 37)(22 52 79 159 38)(23 53 80 160 39)(24 54 81 161 40)(25 180 120 74 102)(26 175 115 75 97)(27 176 116 76 98)(28 177 117 77 99)(29 178 118 78 100)(30 179 119 73 101)(43 86 127 145 64)(44 87 128 146 65)(45 88 129 147 66)(46 89 130 148 61)(47 90 131 149 62)(48 85 132 150 63)
(1 107)(2 108)(3 103)(4 104)(5 105)(6 106)(7 174)(8 169)(9 170)(10 171)(11 172)(12 173)(13 113)(14 114)(15 109)(16 110)(17 111)(18 112)(19 52)(20 53)(21 54)(22 49)(23 50)(24 51)(25 117)(26 118)(27 119)(28 120)(29 115)(30 116)(31 92)(32 93)(33 94)(34 95)(35 96)(36 91)(37 81)(38 82)(39 83)(40 84)(41 79)(42 80)(43 89)(44 90)(45 85)(46 86)(47 87)(48 88)(55 58)(56 59)(57 60)(61 127)(62 128)(63 129)(64 130)(65 131)(66 132)(67 70)(68 71)(69 72)(73 98)(74 99)(75 100)(76 101)(77 102)(78 97)(121 165)(122 166)(123 167)(124 168)(125 163)(126 164)(133 136)(134 137)(135 138)(139 156)(140 151)(141 152)(142 153)(143 154)(144 155)(145 148)(146 149)(147 150)(157 160)(158 161)(159 162)(175 178)(176 179)(177 180)

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

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

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

144 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A ··· 3H 5A 5B 6A ··· 6X 6Y ··· 6BD 10A ··· 10F 15A ··· 15P 30A ··· 30AV order 1 2 2 2 2 2 2 2 3 ··· 3 5 5 6 ··· 6 6 ··· 6 10 ··· 10 15 ··· 15 30 ··· 30 size 1 1 1 1 5 5 5 5 1 ··· 1 2 2 1 ··· 1 5 ··· 5 2 ··· 2 2 ··· 2 2 ··· 2

144 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C3 C6 C6 D5 D10 C3×D5 C6×D5 kernel D5×C62 D5×C3×C6 C6×C30 D5×C2×C6 C6×D5 C2×C30 C62 C3×C6 C2×C6 C6 # reps 1 6 1 8 48 8 2 6 16 48

Matrix representation of D5×C62 in GL3(𝔽31) generated by

 25 0 0 0 30 0 0 0 30
,
 26 0 0 0 6 0 0 0 6
,
 1 0 0 0 30 14 0 30 13
,
 30 0 0 0 1 0 0 1 30
G:=sub<GL(3,GF(31))| [25,0,0,0,30,0,0,0,30],[26,0,0,0,6,0,0,0,6],[1,0,0,0,30,30,0,14,13],[30,0,0,0,1,1,0,0,30] >;

D5×C62 in GAP, Magma, Sage, TeX

D_5\times C_6^2
% in TeX

G:=Group("D5xC6^2");
// GroupNames label

G:=SmallGroup(360,157);
// by ID

G=gap.SmallGroup(360,157);
# by ID

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

G:=Group<a,b,c,d|a^6=b^6=c^5=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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