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## G = C2×C5⋊D15order 300 = 22·3·52

### Direct product of C2 and C5⋊D15

Aliases: C2×C5⋊D15, C10⋊D15, C52D30, C301D5, C528D6, C156D10, C6⋊(C5⋊D5), (C5×C30)⋊1C2, (C5×C10)⋊3S3, (C5×C15)⋊6C22, C32(C2×C5⋊D5), SmallGroup(300,48)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C15 — C2×C5⋊D15
 Chief series C1 — C5 — C52 — C5×C15 — C5⋊D15 — C2×C5⋊D15
 Lower central C5×C15 — C2×C5⋊D15
 Upper central C1 — C2

Generators and relations for C2×C5⋊D15
G = < a,b,c,d | a2=b5=c15=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 704 in 80 conjugacy classes, 35 normal (9 characteristic)
C1, C2, C2, C3, C22, C5, S3, C6, D5, C10, D6, C15, D10, C52, D15, C30, C5⋊D5, C5×C10, D30, C5×C15, C2×C5⋊D5, C5⋊D15, C5×C30, C2×C5⋊D15
Quotients: C1, C2, C22, S3, D5, D6, D10, D15, C5⋊D5, D30, C2×C5⋊D5, C5⋊D15, C2×C5⋊D15

Smallest permutation representation of C2×C5⋊D15
On 150 points
Generators in S150
(1 87)(2 88)(3 89)(4 90)(5 76)(6 77)(7 78)(8 79)(9 80)(10 81)(11 82)(12 83)(13 84)(14 85)(15 86)(16 146)(17 147)(18 148)(19 149)(20 150)(21 136)(22 137)(23 138)(24 139)(25 140)(26 141)(27 142)(28 143)(29 144)(30 145)(31 54)(32 55)(33 56)(34 57)(35 58)(36 59)(37 60)(38 46)(39 47)(40 48)(41 49)(42 50)(43 51)(44 52)(45 53)(61 103)(62 104)(63 105)(64 91)(65 92)(66 93)(67 94)(68 95)(69 96)(70 97)(71 98)(72 99)(73 100)(74 101)(75 102)(106 129)(107 130)(108 131)(109 132)(110 133)(111 134)(112 135)(113 121)(114 122)(115 123)(116 124)(117 125)(118 126)(119 127)(120 128)
(1 57 26 95 109)(2 58 27 96 110)(3 59 28 97 111)(4 60 29 98 112)(5 46 30 99 113)(6 47 16 100 114)(7 48 17 101 115)(8 49 18 102 116)(9 50 19 103 117)(10 51 20 104 118)(11 52 21 105 119)(12 53 22 91 120)(13 54 23 92 106)(14 55 24 93 107)(15 56 25 94 108)(31 138 65 129 84)(32 139 66 130 85)(33 140 67 131 86)(34 141 68 132 87)(35 142 69 133 88)(36 143 70 134 89)(37 144 71 135 90)(38 145 72 121 76)(39 146 73 122 77)(40 147 74 123 78)(41 148 75 124 79)(42 149 61 125 80)(43 150 62 126 81)(44 136 63 127 82)(45 137 64 128 83)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 104)(17 103)(18 102)(19 101)(20 100)(21 99)(22 98)(23 97)(24 96)(25 95)(26 94)(27 93)(28 92)(29 91)(30 105)(31 134)(32 133)(33 132)(34 131)(35 130)(36 129)(37 128)(38 127)(39 126)(40 125)(41 124)(42 123)(43 122)(44 121)(45 135)(46 119)(47 118)(48 117)(49 116)(50 115)(51 114)(52 113)(53 112)(54 111)(55 110)(56 109)(57 108)(58 107)(59 106)(60 120)(61 147)(62 146)(63 145)(64 144)(65 143)(66 142)(67 141)(68 140)(69 139)(70 138)(71 137)(72 136)(73 150)(74 149)(75 148)(76 82)(77 81)(78 80)(83 90)(84 89)(85 88)(86 87)

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

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

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

78 conjugacy classes

 class 1 2A 2B 2C 3 5A ··· 5L 6 10A ··· 10L 15A ··· 15X 30A ··· 30X order 1 2 2 2 3 5 ··· 5 6 10 ··· 10 15 ··· 15 30 ··· 30 size 1 1 75 75 2 2 ··· 2 2 2 ··· 2 2 ··· 2 2 ··· 2

78 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 S3 D5 D6 D10 D15 D30 kernel C2×C5⋊D15 C5⋊D15 C5×C30 C5×C10 C30 C52 C15 C10 C5 # reps 1 2 1 1 12 1 12 24 24

Matrix representation of C2×C5⋊D15 in GL4(𝔽31) generated by

 30 0 0 0 0 30 0 0 0 0 30 0 0 0 0 30
,
 30 1 0 0 11 19 0 0 0 0 0 1 0 0 30 18
,
 1 0 0 0 0 1 0 0 0 0 12 4 0 0 27 22
,
 0 13 0 0 12 0 0 0 0 0 20 9 0 0 28 11
G:=sub<GL(4,GF(31))| [30,0,0,0,0,30,0,0,0,0,30,0,0,0,0,30],[30,11,0,0,1,19,0,0,0,0,0,30,0,0,1,18],[1,0,0,0,0,1,0,0,0,0,12,27,0,0,4,22],[0,12,0,0,13,0,0,0,0,0,20,28,0,0,9,11] >;

C2×C5⋊D15 in GAP, Magma, Sage, TeX

C_2\times C_5\rtimes D_{15}
% in TeX

G:=Group("C2xC5:D15");
// GroupNames label

G:=SmallGroup(300,48);
// by ID

G=gap.SmallGroup(300,48);
# by ID

G:=PCGroup([5,-2,-2,-3,-5,-5,122,963,6004]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^5=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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