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## G = (C2×D8).D5order 320 = 26·5

### 3rd non-split extension by C2×D8 of D5 acting via D5/C5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — (C2×D8).D5
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4×Dic5 — D4×Dic5 — (C2×D8).D5
 Lower central C5 — C10 — C2×C20 — (C2×D8).D5
 Upper central C1 — C22 — C2×C4 — C2×D8

Generators and relations for (C2×D8).D5
G = < a,b,c,d,e | a2=b8=c2=d5=1, e2=ab4, ebe-1=ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 470 in 124 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, SD16, C22×C4, C2×D4, C2×Q8, Dic5, C20, C2×C10, C2×C10, D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C52C8, C40, Dic10, C2×Dic5, C2×Dic5, C2×C20, C5×D4, C5×D4, C22×C10, D4.2D4, C2×C52C8, C4×Dic5, C4⋊Dic5, D4.D5, C23.D5, C2×C40, C5×D8, C2×Dic10, C22×Dic5, D4×C10, C20.8Q8, C20.44D4, D4⋊Dic5, C2×D4.D5, D4×Dic5, C20.17D4, C10×D8, (C2×D8).D5
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C4○D8, C8⋊C22, C5⋊D4, C22×D5, D4.2D4, D4×D5, D42D5, C2×C5⋊D4, D8⋊D5, D83D5, Dic5⋊D4, (C2×D8).D5

Smallest permutation representation of (C2×D8).D5
On 160 points
Generators in S160
(1 115)(2 116)(3 117)(4 118)(5 119)(6 120)(7 113)(8 114)(9 93)(10 94)(11 95)(12 96)(13 89)(14 90)(15 91)(16 92)(17 109)(18 110)(19 111)(20 112)(21 105)(22 106)(23 107)(24 108)(25 133)(26 134)(27 135)(28 136)(29 129)(30 130)(31 131)(32 132)(33 102)(34 103)(35 104)(36 97)(37 98)(38 99)(39 100)(40 101)(41 138)(42 139)(43 140)(44 141)(45 142)(46 143)(47 144)(48 137)(49 125)(50 126)(51 127)(52 128)(53 121)(54 122)(55 123)(56 124)(57 88)(58 81)(59 82)(60 83)(61 84)(62 85)(63 86)(64 87)(65 154)(66 155)(67 156)(68 157)(69 158)(70 159)(71 160)(72 153)(73 145)(74 146)(75 147)(76 148)(77 149)(78 150)(79 151)(80 152)
(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)(2 117)(3 116)(4 115)(5 114)(6 113)(7 120)(8 119)(9 96)(10 95)(11 94)(12 93)(13 92)(14 91)(15 90)(16 89)(17 18)(19 24)(20 23)(21 22)(25 32)(26 31)(27 30)(28 29)(33 103)(34 102)(35 101)(36 100)(37 99)(38 98)(39 97)(40 104)(41 44)(42 43)(45 48)(46 47)(49 128)(50 127)(51 126)(52 125)(53 124)(54 123)(55 122)(56 121)(57 62)(58 61)(59 60)(63 64)(65 68)(66 67)(69 72)(70 71)(73 152)(74 151)(75 150)(76 149)(77 148)(78 147)(79 146)(80 145)(81 84)(82 83)(85 88)(86 87)(105 106)(107 112)(108 111)(109 110)(129 136)(130 135)(131 134)(132 133)(137 142)(138 141)(139 140)(143 144)(153 158)(154 157)(155 156)(159 160)
(1 9 97 121 79)(2 10 98 122 80)(3 11 99 123 73)(4 12 100 124 74)(5 13 101 125 75)(6 14 102 126 76)(7 15 103 127 77)(8 16 104 128 78)(17 136 143 66 59)(18 129 144 67 60)(19 130 137 68 61)(20 131 138 69 62)(21 132 139 70 63)(22 133 140 71 64)(23 134 141 72 57)(24 135 142 65 58)(25 43 160 87 106)(26 44 153 88 107)(27 45 154 81 108)(28 46 155 82 109)(29 47 156 83 110)(30 48 157 84 111)(31 41 158 85 112)(32 42 159 86 105)(33 50 148 120 90)(34 51 149 113 91)(35 52 150 114 92)(36 53 151 115 93)(37 54 152 116 94)(38 55 145 117 95)(39 56 146 118 96)(40 49 147 119 89)
(1 154 119 69)(2 66 120 159)(3 156 113 71)(4 68 114 153)(5 158 115 65)(6 70 116 155)(7 160 117 67)(8 72 118 157)(9 45 89 138)(10 143 90 42)(11 47 91 140)(12 137 92 44)(13 41 93 142)(14 139 94 46)(15 43 95 144)(16 141 96 48)(17 50 105 122)(18 127 106 55)(19 52 107 124)(20 121 108 49)(21 54 109 126)(22 123 110 51)(23 56 111 128)(24 125 112 53)(25 38 129 103)(26 100 130 35)(27 40 131 97)(28 102 132 37)(29 34 133 99)(30 104 134 39)(31 36 135 101)(32 98 136 33)(57 146 84 78)(58 75 85 151)(59 148 86 80)(60 77 87 145)(61 150 88 74)(62 79 81 147)(63 152 82 76)(64 73 83 149)

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

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

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

47 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 8A 8B 8C 8D 10A ··· 10F 10G ··· 10N 20A 20B 20C 20D 40A ··· 40H order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 10 ··· 10 20 20 20 20 40 ··· 40 size 1 1 1 1 4 4 8 2 2 10 10 20 20 20 40 2 2 4 4 20 20 2 ··· 2 8 ··· 8 4 4 4 4 4 ··· 4

47 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D5 C4○D4 D10 D10 C4○D8 C5⋊D4 C8⋊C22 D4⋊2D5 D4×D5 D8⋊D5 D8⋊3D5 kernel (C2×D8).D5 C20.8Q8 C20.44D4 D4⋊Dic5 C2×D4.D5 D4×Dic5 C20.17D4 C10×D8 C2×Dic5 C5×D4 C2×D8 C20 C2×C8 C2×D4 C10 D4 C10 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 8 1 2 2 4 4

Matrix representation of (C2×D8).D5 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 40 0 0 0 0 40
,
 0 24 0 0 29 24 0 0 0 0 40 0 0 0 0 1
,
 0 17 0 0 29 0 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 10 0 0 0 0 37
,
 32 0 0 0 0 32 0 0 0 0 0 37 0 0 31 0
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[0,29,0,0,24,24,0,0,0,0,40,0,0,0,0,1],[0,29,0,0,17,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,10,0,0,0,0,37],[32,0,0,0,0,32,0,0,0,0,0,31,0,0,37,0] >;

(C2×D8).D5 in GAP, Magma, Sage, TeX

(C_2\times D_8).D_5
% in TeX

G:=Group("(C2xD8).D5");
// GroupNames label

G:=SmallGroup(320,780);
// by ID

G=gap.SmallGroup(320,780);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,1094,135,570,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^8=c^2=d^5=1,e^2=a*b^4,e*b*e^-1=a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

׿
×
𝔽