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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 [×3], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×2], C2×C4, C2×C4 [×6], D4 [×4], C23 [×2], D5 [×2], C10 [×3], C42, C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×6], C2×C10, C22⋊C8, D4⋊C4 [×2], C4.Q8, C2.D8, C42⋊C2, C4⋊D4, C52C8, C40, C4×D5 [×2], D20 [×4], C2×Dic5, C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, C22×D5, C23.19D4, C2×C52C8, C4×Dic5, C4⋊Dic5, D10⋊C4 [×2], C5×C4⋊C4 [×2], 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 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], 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 67 5 71)(2 68 6 72)(3 69 7 65)(4 70 8 66)(9 99 13 103)(10 100 14 104)(11 101 15 97)(12 102 16 98)(17 95 21 91)(18 96 22 92)(19 89 23 93)(20 90 24 94)(25 33 29 37)(26 34 30 38)(27 35 31 39)(28 36 32 40)(41 119 45 115)(42 120 46 116)(43 113 47 117)(44 114 48 118)(49 131 53 135)(50 132 54 136)(51 133 55 129)(52 134 56 130)(57 126 61 122)(58 127 62 123)(59 128 63 124)(60 121 64 125)(73 83 77 87)(74 84 78 88)(75 85 79 81)(76 86 80 82)(105 142 109 138)(106 143 110 139)(107 144 111 140)(108 137 112 141)(145 156 149 160)(146 157 150 153)(147 158 151 154)(148 159 152 155)
(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)
(1 148 65 157)(2 151 66 160)(3 146 67 155)(4 149 68 158)(5 152 69 153)(6 147 70 156)(7 150 71 159)(8 145 72 154)(9 124 97 57)(10 127 98 60)(11 122 99 63)(12 125 100 58)(13 128 101 61)(14 123 102 64)(15 126 103 59)(16 121 104 62)(17 45 93 113)(18 48 94 116)(19 43 95 119)(20 46 96 114)(21 41 89 117)(22 44 90 120)(23 47 91 115)(24 42 92 118)(25 107 39 142)(26 110 40 137)(27 105 33 140)(28 108 34 143)(29 111 35 138)(30 106 36 141)(31 109 37 144)(32 112 38 139)(49 81 129 73)(50 84 130 76)(51 87 131 79)(52 82 132 74)(53 85 133 77)(54 88 134 80)(55 83 135 75)(56 86 136 78)
(1 77 21 37 57)(2 78 22 38 58)(3 79 23 39 59)(4 80 24 40 60)(5 73 17 33 61)(6 74 18 34 62)(7 75 19 35 63)(8 76 20 36 64)(9 148 53 41 144)(10 149 54 42 137)(11 150 55 43 138)(12 151 56 44 139)(13 152 49 45 140)(14 145 50 46 141)(15 146 51 47 142)(16 147 52 48 143)(25 126 67 87 91)(26 127 68 88 92)(27 128 69 81 93)(28 121 70 82 94)(29 122 71 83 95)(30 123 72 84 96)(31 124 65 85 89)(32 125 66 86 90)(97 157 133 117 109)(98 158 134 118 110)(99 159 135 119 111)(100 160 136 120 112)(101 153 129 113 105)(102 154 130 114 106)(103 155 131 115 107)(104 156 132 116 108)
(1 102)(2 15)(3 104)(4 9)(5 98)(6 11)(7 100)(8 13)(10 69)(12 71)(14 65)(16 67)(17 118)(18 43)(19 120)(20 45)(21 114)(22 47)(23 116)(24 41)(25 52)(26 133)(27 54)(28 135)(29 56)(30 129)(31 50)(32 131)(33 134)(34 55)(35 136)(36 49)(37 130)(38 51)(39 132)(40 53)(42 93)(44 95)(46 89)(48 91)(57 154)(58 146)(59 156)(60 148)(61 158)(62 150)(63 160)(64 152)(66 103)(68 97)(70 99)(72 101)(73 110)(74 138)(75 112)(76 140)(77 106)(78 142)(79 108)(80 144)(81 137)(82 111)(83 139)(84 105)(85 141)(86 107)(87 143)(88 109)(90 115)(92 117)(94 119)(96 113)(121 159)(122 151)(123 153)(124 145)(125 155)(126 147)(127 157)(128 149)

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

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

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

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