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## G = C2×Q8.D15order 480 = 25·3·5

### Direct product of C2 and Q8.D15

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C5×SL2(𝔽3) — C2×Q8.D15
 Chief series C1 — C2 — Q8 — C5×Q8 — C5×SL2(𝔽3) — Q8.D15 — C2×Q8.D15
 Lower central C5×SL2(𝔽3) — C2×Q8.D15
 Upper central C1 — C22

Generators and relations for C2×Q8.D15
G = < a,b,c,d,e | a2=b4=d15=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, dbd-1=c, ebe-1=b-1c, dcd-1=bc, ece-1=b2c, ede-1=d-1 >

Subgroups: 514 in 78 conjugacy classes, 21 normal (15 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C8, C2×C4, Q8, Q8, C10, C10, Dic3, C2×C6, C15, C2×C8, Q16, C2×Q8, C2×Q8, Dic5, C20, C2×C10, SL2(𝔽3), C2×Dic3, C30, C2×Q16, C52C8, Dic10, C2×Dic5, C2×C20, C5×Q8, C5×Q8, CSU2(𝔽3), C2×SL2(𝔽3), Dic15, C2×C30, C2×C52C8, C5⋊Q16, C2×Dic10, Q8×C10, C2×CSU2(𝔽3), C5×SL2(𝔽3), C2×Dic15, C2×C5⋊Q16, Q8.D15, C10×SL2(𝔽3), C2×Q8.D15
Quotients: C1, C2, C22, S3, D5, D6, D10, S4, D15, CSU2(𝔽3), C2×S4, D30, C2×CSU2(𝔽3), C5⋊S4, Q8.D15, C2×C5⋊S4, C2×Q8.D15

Smallest permutation representation of C2×Q8.D15
On 160 points
Generators in S160
(1 25)(2 21)(3 22)(4 23)(5 24)(6 20)(7 16)(8 17)(9 18)(10 19)(11 27)(12 28)(13 29)(14 30)(15 26)(31 40)(32 36)(33 37)(34 38)(35 39)(41 158)(42 159)(43 160)(44 146)(45 147)(46 148)(47 149)(48 150)(49 151)(50 152)(51 153)(52 154)(53 155)(54 156)(55 157)(56 131)(57 132)(58 133)(59 134)(60 135)(61 136)(62 137)(63 138)(64 139)(65 140)(66 141)(67 142)(68 143)(69 144)(70 145)(71 118)(72 119)(73 120)(74 121)(75 122)(76 123)(77 124)(78 125)(79 126)(80 127)(81 128)(82 129)(83 130)(84 116)(85 117)(86 113)(87 114)(88 115)(89 101)(90 102)(91 103)(92 104)(93 105)(94 106)(95 107)(96 108)(97 109)(98 110)(99 111)(100 112)
(1 108 15 152)(2 114 11 158)(3 105 12 149)(4 111 13 155)(5 102 14 146)(6 66 39 76)(7 57 40 82)(8 63 36 73)(9 69 37 79)(10 60 38 85)(16 132 31 129)(17 138 32 120)(18 144 33 126)(19 135 34 117)(20 141 35 123)(21 87 27 41)(22 93 28 47)(23 99 29 53)(24 90 30 44)(25 96 26 50)(42 52 88 98)(43 94 89 48)(45 55 91 86)(46 97 92 51)(49 100 95 54)(56 71 81 61)(58 68 83 78)(59 74 84 64)(62 77 72 67)(65 80 75 70)(101 150 160 106)(103 113 147 157)(104 153 148 109)(107 156 151 112)(110 159 154 115)(116 139 134 121)(118 128 136 131)(119 142 137 124)(122 145 140 127)(125 133 143 130)
(1 113 15 157)(2 104 11 148)(3 110 12 154)(4 101 13 160)(5 107 14 151)(6 56 39 81)(7 62 40 72)(8 68 36 78)(9 59 37 84)(10 65 38 75)(16 137 31 119)(17 143 32 125)(18 134 33 116)(19 140 34 122)(20 131 35 128)(21 92 27 46)(22 98 28 52)(23 89 29 43)(24 95 30 49)(25 86 26 55)(41 51 87 97)(42 93 88 47)(44 54 90 100)(45 96 91 50)(48 99 94 53)(57 67 82 77)(58 73 83 63)(60 70 85 80)(61 76 71 66)(64 79 74 69)(102 112 146 156)(103 152 147 108)(105 115 149 159)(106 155 150 111)(109 158 153 114)(117 127 135 145)(118 141 136 123)(120 130 138 133)(121 144 139 126)(124 132 142 129)
(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 35 15 20)(2 34 11 19)(3 33 12 18)(4 32 13 17)(5 31 14 16)(6 25 39 26)(7 24 40 30)(8 23 36 29)(9 22 37 28)(10 21 38 27)(41 80 87 70)(42 79 88 69)(43 78 89 68)(44 77 90 67)(45 76 91 66)(46 75 92 65)(47 74 93 64)(48 73 94 63)(49 72 95 62)(50 71 96 61)(51 85 97 60)(52 84 98 59)(53 83 99 58)(54 82 100 57)(55 81 86 56)(101 143 160 125)(102 142 146 124)(103 141 147 123)(104 140 148 122)(105 139 149 121)(106 138 150 120)(107 137 151 119)(108 136 152 118)(109 135 153 117)(110 134 154 116)(111 133 155 130)(112 132 156 129)(113 131 157 128)(114 145 158 127)(115 144 159 126)

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

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

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

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 3 3 4 4 6 6 type + + + + + + + + - + + + - - + + image C1 C2 C2 S3 D5 D6 D10 D15 CSU2(𝔽3) D30 S4 C2×S4 CSU2(𝔽3) Q8.D15 C5⋊S4 C2×C5⋊S4 kernel C2×Q8.D15 Q8.D15 C10×SL2(𝔽3) Q8×C10 C2×SL2(𝔽3) C5×Q8 SL2(𝔽3) C2×Q8 C10 Q8 C2×C10 C10 C10 C2 C22 C2 # reps 1 2 1 1 2 1 2 4 4 4 2 2 2 12 2 2

Matrix representation of C2×Q8.D15 in GL4(𝔽241) generated by

 240 0 0 0 0 240 0 0 0 0 240 0 0 0 0 240
,
 1 0 0 0 0 1 0 0 0 0 4 205 0 0 208 237
,
 1 0 0 0 0 1 0 0 0 0 32 5 0 0 36 209
,
 66 31 0 0 117 128 0 0 0 0 0 240 0 0 1 240
,
 56 188 0 0 141 185 0 0 0 0 31 36 0 0 67 210
G:=sub<GL(4,GF(241))| [240,0,0,0,0,240,0,0,0,0,240,0,0,0,0,240],[1,0,0,0,0,1,0,0,0,0,4,208,0,0,205,237],[1,0,0,0,0,1,0,0,0,0,32,36,0,0,5,209],[66,117,0,0,31,128,0,0,0,0,0,1,0,0,240,240],[56,141,0,0,188,185,0,0,0,0,31,67,0,0,36,210] >;

C2×Q8.D15 in GAP, Magma, Sage, TeX

C_2\times Q_8.D_{15}
% in TeX

G:=Group("C2xQ8.D15");
// GroupNames label

G:=SmallGroup(480,1027);
// by ID

G=gap.SmallGroup(480,1027);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,2,-2,1680,170,1347,4204,3168,172,2525,1909,285,124]);
// Polycyclic

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

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