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G = C4.Q8⋊D5order 320 = 26·5

9th semidirect product of C4.Q8 and D5 acting via D5/C5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4.Q89D5, C4⋊C4.40D10, C4⋊D20.7C2, (C2×C8).139D10, D205C432C2, D101C831C2, D206C417C2, C4.74(C4○D20), C20.31(C4○D4), C10.56(C4○D8), C10.D816C2, (C22×D5).33D4, C22.218(D4×D5), C2.23(D40⋊C2), C10.71(C8⋊C22), (C2×C20).282C23, (C2×C40).286C22, C4.26(Q82D5), (C2×Dic5).219D4, (C2×D20).80C22, C54(C23.19D4), C4⋊Dic5.112C22, C2.23(SD163D5), C2.13(D10.13D4), C10.43(C22.D4), C4⋊C47D56C2, (C5×C4.Q8)⋊17C2, (C2×C4×D5).38C22, (C2×C10).287(C2×D4), (C5×C4⋊C4).75C22, (C2×C52C8).59C22, (C2×C4).385(C22×D5), SmallGroup(320,493)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C4.Q8⋊D5
C1C5C10C2×C10C2×C20C2×C4×D5C4⋊C47D5 — C4.Q8⋊D5
C5C10C2×C20 — C4.Q8⋊D5
C1C22C2×C4C4.Q8

Generators and relations for C4.Q8⋊D5
 G = < a,b,c,d,e | a4=d5=e2=1, b4=a2, c2=a-1b2, ab=ba, cac-1=a-1, ad=da, ae=ea, cbc-1=b3, bd=db, ebe=a-1b3, cd=dc, ece=a2c, ede=d-1 >

Subgroups: 526 in 106 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, C22⋊C8, D4⋊C4, C4.Q8, C2.D8, C42⋊C2, C4⋊D4, C52C8, C40, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C23.19D4, C2×C52C8, C4×Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C40, C2×C4×D5, C2×D20, C2×D20, C10.D8, D206C4, D101C8, D205C4, C5×C4.Q8, C4⋊C47D5, C4⋊D20, C4.Q8⋊D5
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C22.D4, C4○D8, C8⋊C22, C22×D5, C23.19D4, C4○D20, D4×D5, Q82D5, D10.13D4, D40⋊C2, SD163D5, C4.Q8⋊D5

Smallest permutation representation of C4.Q8⋊D5
On 160 points
Generators in S160
(1 73 5 77)(2 74 6 78)(3 75 7 79)(4 76 8 80)(9 150 13 146)(10 151 14 147)(11 152 15 148)(12 145 16 149)(17 92 21 96)(18 93 22 89)(19 94 23 90)(20 95 24 91)(25 34 29 38)(26 35 30 39)(27 36 31 40)(28 37 32 33)(41 136 45 132)(42 129 46 133)(43 130 47 134)(44 131 48 135)(49 103 53 99)(50 104 54 100)(51 97 55 101)(52 98 56 102)(57 87 61 83)(58 88 62 84)(59 81 63 85)(60 82 64 86)(65 122 69 126)(66 123 70 127)(67 124 71 128)(68 125 72 121)(105 138 109 142)(106 139 110 143)(107 140 111 144)(108 141 112 137)(113 154 117 158)(114 155 118 159)(115 156 119 160)(116 157 120 153)
(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)
(1 118 79 157)(2 113 80 160)(3 116 73 155)(4 119 74 158)(5 114 75 153)(6 117 76 156)(7 120 77 159)(8 115 78 154)(9 124 148 69)(10 127 149 72)(11 122 150 67)(12 125 151 70)(13 128 152 65)(14 123 145 68)(15 126 146 71)(16 121 147 66)(17 46 90 131)(18 41 91 134)(19 44 92 129)(20 47 93 132)(21 42 94 135)(22 45 95 130)(23 48 96 133)(24 43 89 136)(25 110 40 141)(26 105 33 144)(27 108 34 139)(28 111 35 142)(29 106 36 137)(30 109 37 140)(31 112 38 143)(32 107 39 138)(49 81 101 61)(50 84 102 64)(51 87 103 59)(52 82 104 62)(53 85 97 57)(54 88 98 60)(55 83 99 63)(56 86 100 58)
(1 57 67 37 21)(2 58 68 38 22)(3 59 69 39 23)(4 60 70 40 24)(5 61 71 33 17)(6 62 72 34 18)(7 63 65 35 19)(8 64 66 36 20)(9 138 48 116 51)(10 139 41 117 52)(11 140 42 118 53)(12 141 43 119 54)(13 142 44 120 55)(14 143 45 113 56)(15 144 46 114 49)(16 137 47 115 50)(25 89 74 88 125)(26 90 75 81 126)(27 91 76 82 127)(28 92 77 83 128)(29 93 78 84 121)(30 94 79 85 122)(31 95 80 86 123)(32 96 73 87 124)(97 150 109 135 157)(98 151 110 136 158)(99 152 111 129 159)(100 145 112 130 160)(101 146 105 131 153)(102 147 106 132 154)(103 148 107 133 155)(104 149 108 134 156)
(1 132)(2 48)(3 134)(4 42)(5 136)(6 44)(7 130)(8 46)(9 68)(10 124)(11 70)(12 126)(13 72)(14 128)(15 66)(16 122)(17 158)(18 120)(19 160)(20 114)(21 154)(22 116)(23 156)(24 118)(25 97)(26 54)(27 99)(28 56)(29 101)(30 50)(31 103)(32 52)(33 98)(34 55)(35 100)(36 49)(37 102)(38 51)(39 104)(40 53)(41 73)(43 75)(45 77)(47 79)(57 106)(58 138)(59 108)(60 140)(61 110)(62 142)(63 112)(64 144)(65 145)(67 147)(69 149)(71 151)(74 135)(76 129)(78 131)(80 133)(81 141)(82 111)(83 143)(84 105)(85 137)(86 107)(87 139)(88 109)(89 157)(90 119)(91 159)(92 113)(93 153)(94 115)(95 155)(96 117)(121 146)(123 148)(125 150)(127 152)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12222244444444455888810···102020202020···2040···40
size111120402244810102020224420202···244448···84···4

47 irreducible representations

dim111111112222222244444
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D5C4○D4D10D10C4○D8C4○D20C8⋊C22Q82D5D4×D5D40⋊C2SD163D5
kernelC4.Q8⋊D5C10.D8D206C4D101C8D205C4C5×C4.Q8C4⋊C47D5C4⋊D20C2×Dic5C22×D5C4.Q8C20C4⋊C4C2×C8C10C4C10C4C22C2C2
# reps111111111124424812244

Matrix representation of C4.Q8⋊D5 in GL4(𝔽41) generated by

403900
1100
00400
00040
,
303000
26000
00236
003518
,
02400
12000
00320
00032
,
1000
0100
0001
00406
,
322300
9900
002839
00213
G:=sub<GL(4,GF(41))| [40,1,0,0,39,1,0,0,0,0,40,0,0,0,0,40],[30,26,0,0,30,0,0,0,0,0,23,35,0,0,6,18],[0,12,0,0,24,0,0,0,0,0,32,0,0,0,0,32],[1,0,0,0,0,1,0,0,0,0,0,40,0,0,1,6],[32,9,0,0,23,9,0,0,0,0,28,2,0,0,39,13] >;

C4.Q8⋊D5 in GAP, Magma, Sage, TeX

C_4.Q_8\rtimes D_5
% in TeX

G:=Group("C4.Q8:D5");
// GroupNames label

G:=SmallGroup(320,493);
// by ID

G=gap.SmallGroup(320,493);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,64,254,219,100,851,102,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=d^5=e^2=1,b^4=a^2,c^2=a^-1*b^2,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,a*e=e*a,c*b*c^-1=b^3,b*d=d*b,e*b*e=a^-1*b^3,c*d=d*c,e*c*e=a^2*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽