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## G = (C2×D4).8F5order 320 = 26·5

### 5th non-split extension by C2×D4 of F5 acting via F5/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — (C2×D4).8F5
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — C2×D5⋊C8 — (C2×D4).8F5
 Lower central C5 — C2×C10 — (C2×D4).8F5
 Upper central C1 — C22 — C2×D4

Generators and relations for (C2×D4).8F5
G = < a,b,c,d,e | a2=b4=c2=d5=1, e4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=ab2c, ede-1=d3 >

Subgroups: 586 in 158 conjugacy classes, 52 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×4], C22, C22 [×10], C5, C8 [×4], C2×C4, C2×C4 [×11], D4 [×6], Q8 [×2], C23 [×2], C23, D5 [×2], C10, C10 [×2], C10 [×2], C2×C8 [×6], M4(2) [×2], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×4], Dic5 [×2], Dic5 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C10 [×6], C22⋊C8 [×4], C22×C8, C2×M4(2), C2×C4○D4, C5⋊C8 [×4], Dic10 [×2], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×Dic5 [×4], C5⋊D4 [×4], C2×C20, C5×D4 [×2], C22×D5, C22×C10 [×2], (C22×C8)⋊C2, D5⋊C8 [×2], C4.F5 [×2], C2×C5⋊C8 [×4], C2×Dic10, C2×C4×D5, D42D5 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×2], D4×C10, C23.2F5 [×4], C2×D5⋊C8, C2×C4.F5, C2×D42D5, (C2×D4).8F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], F5, C2×C22⋊C4, C8○D4 [×2], C2×F5 [×3], (C22×C8)⋊C2, C22⋊F5 [×2], C22×F5, D4.F5 [×2], C2×C22⋊F5, (C2×D4).8F5

Smallest permutation representation of (C2×D4).8F5
On 160 points
Generators in S160
(1 61)(2 62)(3 63)(4 64)(5 57)(6 58)(7 59)(8 60)(9 20)(10 21)(11 22)(12 23)(13 24)(14 17)(15 18)(16 19)(25 121)(26 122)(27 123)(28 124)(29 125)(30 126)(31 127)(32 128)(33 100)(34 101)(35 102)(36 103)(37 104)(38 97)(39 98)(40 99)(41 137)(42 138)(43 139)(44 140)(45 141)(46 142)(47 143)(48 144)(49 110)(50 111)(51 112)(52 105)(53 106)(54 107)(55 108)(56 109)(65 81)(66 82)(67 83)(68 84)(69 85)(70 86)(71 87)(72 88)(73 149)(74 150)(75 151)(76 152)(77 145)(78 146)(79 147)(80 148)(89 134)(90 135)(91 136)(92 129)(93 130)(94 131)(95 132)(96 133)(113 155)(114 156)(115 157)(116 158)(117 159)(118 160)(119 153)(120 154)
(1 63 5 59)(2 64 6 60)(3 57 7 61)(4 58 8 62)(9 104 13 100)(10 97 14 101)(11 98 15 102)(12 99 16 103)(17 34 21 38)(18 35 22 39)(19 36 23 40)(20 37 24 33)(25 54 29 50)(26 55 30 51)(27 56 31 52)(28 49 32 53)(41 136 45 132)(42 129 46 133)(43 130 47 134)(44 131 48 135)(65 87 69 83)(66 88 70 84)(67 81 71 85)(68 82 72 86)(73 117 77 113)(74 118 78 114)(75 119 79 115)(76 120 80 116)(89 139 93 143)(90 140 94 144)(91 141 95 137)(92 142 96 138)(105 123 109 127)(106 124 110 128)(107 125 111 121)(108 126 112 122)(145 155 149 159)(146 156 150 160)(147 157 151 153)(148 158 152 154)
(1 69)(2 82)(3 71)(4 84)(5 65)(6 86)(7 67)(8 88)(9 43)(10 144)(11 45)(12 138)(13 47)(14 140)(15 41)(16 142)(17 44)(18 137)(19 46)(20 139)(21 48)(22 141)(23 42)(24 143)(25 157)(26 120)(27 159)(28 114)(29 153)(30 116)(31 155)(32 118)(33 93)(34 135)(35 95)(36 129)(37 89)(38 131)(39 91)(40 133)(49 78)(50 151)(51 80)(52 145)(53 74)(54 147)(55 76)(56 149)(57 81)(58 70)(59 83)(60 72)(61 85)(62 66)(63 87)(64 68)(73 109)(75 111)(77 105)(79 107)(90 101)(92 103)(94 97)(96 99)(98 136)(100 130)(102 132)(104 134)(106 150)(108 152)(110 146)(112 148)(113 127)(115 121)(117 123)(119 125)(122 154)(124 156)(126 158)(128 160)
(1 147 132 143 113)(2 144 148 114 133)(3 115 137 134 149)(4 135 116 150 138)(5 151 136 139 117)(6 140 152 118 129)(7 119 141 130 145)(8 131 120 146 142)(9 27 81 111 39)(10 112 28 40 82)(11 33 105 83 29)(12 84 34 30 106)(13 31 85 107 35)(14 108 32 36 86)(15 37 109 87 25)(16 88 38 26 110)(17 55 128 103 70)(18 104 56 71 121)(19 72 97 122 49)(20 123 65 50 98)(21 51 124 99 66)(22 100 52 67 125)(23 68 101 126 53)(24 127 69 54 102)(41 89 73 63 157)(42 64 90 158 74)(43 159 57 75 91)(44 76 160 92 58)(45 93 77 59 153)(46 60 94 154 78)(47 155 61 79 95)(48 80 156 96 62)
(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)

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 5 8A ··· 8H 8I 8J 8K 8L 10A 10B 10C 10D 10E 10F 10G 20A 20B order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 5 8 ··· 8 8 8 8 8 10 10 10 10 10 10 10 20 20 size 1 1 1 1 4 4 10 10 2 2 5 5 5 5 20 20 4 10 ··· 10 20 20 20 20 4 4 4 8 8 8 8 8 8

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 4 4 4 4 8 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C4 C4 C4 D4 C8○D4 F5 C2×F5 C2×F5 C22⋊F5 D4.F5 kernel (C2×D4).8F5 C23.2F5 C2×D5⋊C8 C2×C4.F5 C2×D4⋊2D5 C2×Dic10 C2×C5⋊D4 D4×C10 C4×D5 C10 C2×D4 C2×C4 C23 C4 C2 # reps 1 4 1 1 1 2 4 2 4 8 1 1 2 4 2

Matrix representation of (C2×D4).8F5 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 32 0 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 9 0 0 0 0 32 0 0 0 0 0 0 0 22 38 0 3 0 0 0 19 38 3 0 0 3 38 19 0 0 0 3 0 38 22
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 40 0 0 1 0 0 40 0 0 0 1 0 40 0 0 0 0 1 40
,
 38 0 0 0 0 0 0 38 0 0 0 0 0 0 13 28 20 0 0 0 33 28 0 13 0 0 13 0 28 33 0 0 0 20 28 13

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,32,0,0,0,0,9,0,0,0,0,0,0,0,22,0,3,3,0,0,38,19,38,0,0,0,0,38,19,38,0,0,3,3,0,22],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[38,0,0,0,0,0,0,38,0,0,0,0,0,0,13,33,13,0,0,0,28,28,0,20,0,0,20,0,28,28,0,0,0,13,33,13] >;

(C2×D4).8F5 in GAP, Magma, Sage, TeX

(C_2\times D_4)._8F_5
% in TeX

G:=Group("(C2xD4).8F5");
// GroupNames label

G:=SmallGroup(320,1114);
// by ID

G=gap.SmallGroup(320,1114);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,387,136,6278,1595]);
// Polycyclic

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

׿
×
𝔽