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## G = C6×C5⋊D5order 300 = 22·3·52

### Direct product of C6 and C5⋊D5

Aliases: C6×C5⋊D5, C302D5, C157D10, C10⋊(C3×D5), C52(C6×D5), (C5×C10)⋊4C6, (C5×C30)⋊4C2, C525(C2×C6), (C5×C15)⋊9C22, SmallGroup(300,45)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C6×C5⋊D5
 Chief series C1 — C5 — C52 — C5×C15 — C3×C5⋊D5 — C6×C5⋊D5
 Lower central C52 — C6×C5⋊D5
 Upper central C1 — C6

Generators and relations for C6×C5⋊D5
G = < a,b,c,d | a6=b5=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 368 in 80 conjugacy classes, 38 normal (10 characteristic)
C1, C2, C2, C3, C22, C5, C6, C6, D5, C10, C2×C6, C15, D10, C52, C3×D5, C30, C5⋊D5, C5×C10, C6×D5, C5×C15, C2×C5⋊D5, C3×C5⋊D5, C5×C30, C6×C5⋊D5
Quotients: C1, C2, C3, C22, C6, D5, C2×C6, D10, C3×D5, C5⋊D5, C6×D5, C2×C5⋊D5, C3×C5⋊D5, C6×C5⋊D5

Smallest permutation representation of C6×C5⋊D5
On 150 points
Generators in S150
(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)
(1 103 86 135 118)(2 104 87 136 119)(3 105 88 137 120)(4 106 89 138 115)(5 107 90 133 116)(6 108 85 134 117)(7 95 141 75 36)(8 96 142 76 31)(9 91 143 77 32)(10 92 144 78 33)(11 93 139 73 34)(12 94 140 74 35)(13 68 98 59 42)(14 69 99 60 37)(15 70 100 55 38)(16 71 101 56 39)(17 72 102 57 40)(18 67 97 58 41)(19 28 81 62 48)(20 29 82 63 43)(21 30 83 64 44)(22 25 84 65 45)(23 26 79 66 46)(24 27 80 61 47)(49 146 131 114 124)(50 147 132 109 125)(51 148 127 110 126)(52 149 128 111 121)(53 150 129 112 122)(54 145 130 113 123)
(1 72 111 9 80)(2 67 112 10 81)(3 68 113 11 82)(4 69 114 12 83)(5 70 109 7 84)(6 71 110 8 79)(13 130 34 29 120)(14 131 35 30 115)(15 132 36 25 116)(16 127 31 26 117)(17 128 32 27 118)(18 129 33 28 119)(19 136 41 150 78)(20 137 42 145 73)(21 138 37 146 74)(22 133 38 147 75)(23 134 39 148 76)(24 135 40 149 77)(43 88 59 54 139)(44 89 60 49 140)(45 90 55 50 141)(46 85 56 51 142)(47 86 57 52 143)(48 87 58 53 144)(61 103 102 121 91)(62 104 97 122 92)(63 105 98 123 93)(64 106 99 124 94)(65 107 100 125 95)(66 108 101 126 96)
(1 80)(2 81)(3 82)(4 83)(5 84)(6 79)(7 70)(8 71)(9 72)(10 67)(11 68)(12 69)(13 93)(14 94)(15 95)(16 96)(17 91)(18 92)(19 87)(20 88)(21 89)(22 90)(23 85)(24 86)(25 107)(26 108)(27 103)(28 104)(29 105)(30 106)(31 101)(32 102)(33 97)(34 98)(35 99)(36 100)(37 140)(38 141)(39 142)(40 143)(41 144)(42 139)(43 137)(44 138)(45 133)(46 134)(47 135)(48 136)(49 146)(50 147)(51 148)(52 149)(53 150)(54 145)(55 75)(56 76)(57 77)(58 78)(59 73)(60 74)(61 118)(62 119)(63 120)(64 115)(65 116)(66 117)(121 128)(122 129)(123 130)(124 131)(125 132)(126 127)

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 3A 3B 5A ··· 5L 6A 6B 6C 6D 6E 6F 10A ··· 10L 15A ··· 15X 30A ··· 30X order 1 2 2 2 3 3 5 ··· 5 6 6 6 6 6 6 10 ··· 10 15 ··· 15 30 ··· 30 size 1 1 25 25 1 1 2 ··· 2 1 1 25 25 25 25 2 ··· 2 2 ··· 2 2 ··· 2

84 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 C6×C5⋊D5 C3×C5⋊D5 C5×C30 C2×C5⋊D5 C5⋊D5 C5×C10 C30 C15 C10 C5 # reps 1 2 1 2 4 2 12 12 24 24

Matrix representation of C6×C5⋊D5 in GL4(𝔽31) generated by

 30 0 0 0 0 30 0 0 0 0 5 0 0 0 0 5
,
 1 0 0 0 0 1 0 0 0 0 30 1 0 0 17 13
,
 0 1 0 0 30 18 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 18 1 0 0 18 13
G:=sub<GL(4,GF(31))| [30,0,0,0,0,30,0,0,0,0,5,0,0,0,0,5],[1,0,0,0,0,1,0,0,0,0,30,17,0,0,1,13],[0,30,0,0,1,18,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,18,18,0,0,1,13] >;

C6×C5⋊D5 in GAP, Magma, Sage, TeX

C_6\times C_5\rtimes D_5
% in TeX

G:=Group("C6xC5:D5");
// GroupNames label

G:=SmallGroup(300,45);
// by ID

G=gap.SmallGroup(300,45);
# by ID

G:=PCGroup([5,-2,-2,-3,-5,-5,963,6004]);
// Polycyclic

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