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## G = Q8⋊Dic15order 480 = 25·3·5

### The semidirect product of Q8 and Dic15 acting via Dic15/C10=S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C5×SL2(𝔽3) — Q8⋊Dic15
 Chief series C1 — C2 — Q8 — C5×Q8 — C5×SL2(𝔽3) — C10×SL2(𝔽3) — Q8⋊Dic15
 Lower central C5×SL2(𝔽3) — Q8⋊Dic15
 Upper central C1 — C22

Generators and relations for Q8⋊Dic15
G = < a,b,c,d | a4=c30=1, b2=a2, d2=c15, bab-1=dbd-1=a-1, cac-1=b, dad-1=a2b, cbc-1=ab, dcd-1=c-1 >

Smallest permutation representation of Q8⋊Dic15
On 160 points
Generators in S160
(1 124 30 62)(2 115 21 53)(3 106 22 44)(4 127 23 65)(5 118 24 56)(6 109 25 47)(7 130 26 68)(8 121 27 59)(9 112 28 50)(10 103 29 41)(11 133 36 97)(12 154 37 88)(13 145 38 79)(14 136 39 100)(15 157 40 91)(16 148 31 82)(17 139 32 73)(18 160 33 94)(19 151 34 85)(20 142 35 76)(42 52 104 114)(43 125 105 63)(45 55 107 117)(46 128 108 66)(48 58 110 120)(49 101 111 69)(51 61 113 123)(54 64 116 126)(57 67 119 129)(60 70 122 102)(71 81 137 147)(72 158 138 92)(74 84 140 150)(75 131 141 95)(77 87 143 153)(78 134 144 98)(80 90 146 156)(83 93 149 159)(86 96 152 132)(89 99 155 135)
(1 114 30 52)(2 105 21 43)(3 126 22 64)(4 117 23 55)(5 108 24 46)(6 129 25 67)(7 120 26 58)(8 111 27 49)(9 102 28 70)(10 123 29 61)(11 153 36 87)(12 144 37 78)(13 135 38 99)(14 156 39 90)(15 147 40 81)(16 138 31 72)(17 159 32 93)(18 150 33 84)(19 141 34 75)(20 132 35 96)(41 51 103 113)(42 124 104 62)(44 54 106 116)(45 127 107 65)(47 57 109 119)(48 130 110 68)(50 60 112 122)(53 63 115 125)(56 66 118 128)(59 69 121 101)(71 157 137 91)(73 83 139 149)(74 160 140 94)(76 86 142 152)(77 133 143 97)(79 89 145 155)(80 136 146 100)(82 92 148 158)(85 95 151 131)(88 98 154 134)
(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)
(1 20 6 15)(2 19 7 14)(3 18 8 13)(4 17 9 12)(5 16 10 11)(21 34 26 39)(22 33 27 38)(23 32 28 37)(24 31 29 36)(25 40 30 35)(41 153 56 138)(42 152 57 137)(43 151 58 136)(44 150 59 135)(45 149 60 134)(46 148 61 133)(47 147 62 132)(48 146 63 131)(49 145 64 160)(50 144 65 159)(51 143 66 158)(52 142 67 157)(53 141 68 156)(54 140 69 155)(55 139 70 154)(71 104 86 119)(72 103 87 118)(73 102 88 117)(74 101 89 116)(75 130 90 115)(76 129 91 114)(77 128 92 113)(78 127 93 112)(79 126 94 111)(80 125 95 110)(81 124 96 109)(82 123 97 108)(83 122 98 107)(84 121 99 106)(85 120 100 105)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 5A 5B 6A 6B 6C 8A 8B 8C 8D 10A ··· 10F 15A 15B 15C 15D 20A 20B 20C 20D 30A ··· 30L order 1 2 2 2 3 4 4 4 4 5 5 6 6 6 8 8 8 8 10 ··· 10 15 15 15 15 20 20 20 20 30 ··· 30 size 1 1 1 1 8 6 6 60 60 2 2 8 8 8 30 30 30 30 2 ··· 2 8 8 8 8 12 12 12 12 8 ··· 8

44 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 2 2 3 3 4 4 4 4 6 6 type + + + + - - + - - + - + - + + - image C1 C2 C4 S3 D5 Dic3 Dic5 D15 CSU2(𝔽3) GL2(𝔽3) Dic15 S4 A4⋊C4 CSU2(𝔽3) GL2(𝔽3) Q8.D15 Q8⋊D15 C5⋊S4 A4⋊Dic5 kernel Q8⋊Dic15 C10×SL2(𝔽3) C5×SL2(𝔽3) Q8×C10 C2×SL2(𝔽3) C5×Q8 SL2(𝔽3) C2×Q8 C10 C10 Q8 C2×C10 C10 C10 C10 C2 C2 C22 C2 # reps 1 1 2 1 2 1 2 4 2 2 4 2 2 1 1 6 6 2 2

Matrix representation of Q8⋊Dic15 in GL4(𝔽241) generated by

 1 0 0 0 0 1 0 0 0 0 150 193 0 0 42 91
,
 1 0 0 0 0 1 0 0 0 0 49 199 0 0 149 192
,
 113 31 0 0 117 175 0 0 0 0 1 1 0 0 240 0
,
 166 127 0 0 117 75 0 0 0 0 147 175 0 0 28 94
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,150,42,0,0,193,91],[1,0,0,0,0,1,0,0,0,0,49,149,0,0,199,192],[113,117,0,0,31,175,0,0,0,0,1,240,0,0,1,0],[166,117,0,0,127,75,0,0,0,0,147,28,0,0,175,94] >;

Q8⋊Dic15 in GAP, Magma, Sage, TeX

Q_8\rtimes {\rm Dic}_{15}
% in TeX

G:=Group("Q8:Dic15");
// GroupNames label

G:=SmallGroup(480,260);
// by ID

G=gap.SmallGroup(480,260);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,2,-2,14,170,1347,4204,3168,172,2525,1909,285,124]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^30=1,b^2=a^2,d^2=c^15,b*a*b^-1=d*b*d^-1=a^-1,c*a*c^-1=b,d*a*d^-1=a^2*b,c*b*c^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations

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