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## G = C5×D4.Q8order 320 = 26·5

### Direct product of C5 and D4.Q8

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C5×D4.Q8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C20 — C5×C4⋊C4 — C5×C42.C2 — C5×D4.Q8
 Lower central C1 — C2 — C2×C4 — C5×D4.Q8
 Upper central C1 — C2×C10 — C4×C20 — C5×D4.Q8

Generators and relations for C5×D4.Q8
G = < a,b,c,d,e | a5=b4=c2=d4=1, e2=b2d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=b2d-1 >

Subgroups: 186 in 102 conjugacy classes, 54 normal (50 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C20, C20, C2×C10, C2×C10, D4⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×D4, C42.C2, C40, C2×C20, C2×C20, C5×D4, C5×D4, C22×C10, D4.Q8, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C22×C20, D4×C10, C5×D4⋊C4, C5×C4⋊C8, C5×C4.Q8, C5×C2.D8, D4×C20, C5×C42.C2, C5×D4.Q8
Quotients: C1, C2, C22, C5, D4, Q8, C23, C10, C2×D4, C2×Q8, C4○D4, C2×C10, C22⋊Q8, C4○D8, C8⋊C22, C5×D4, C5×Q8, C22×C10, D4.Q8, D4×C10, Q8×C10, C5×C4○D4, C5×C22⋊Q8, C5×C4○D8, C5×C8⋊C22, C5×D4.Q8

Smallest permutation representation of C5×D4.Q8
On 160 points
Generators in S160
(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 37 12 66)(2 38 13 67)(3 39 14 68)(4 40 15 69)(5 36 11 70)(6 127 160 18)(7 128 156 19)(8 129 157 20)(9 130 158 16)(10 126 159 17)(21 35 142 151)(22 31 143 152)(23 32 144 153)(24 33 145 154)(25 34 141 155)(26 56 42 51)(27 57 43 52)(28 58 44 53)(29 59 45 54)(30 60 41 55)(46 108 91 79)(47 109 92 80)(48 110 93 76)(49 106 94 77)(50 107 95 78)(61 97 83 75)(62 98 84 71)(63 99 85 72)(64 100 81 73)(65 96 82 74)(86 119 131 148)(87 120 132 149)(88 116 133 150)(89 117 134 146)(90 118 135 147)(101 115 123 137)(102 111 124 138)(103 112 125 139)(104 113 121 140)(105 114 122 136)
(1 55)(2 51)(3 52)(4 53)(5 54)(6 151)(7 152)(8 153)(9 154)(10 155)(11 59)(12 60)(13 56)(14 57)(15 58)(16 24)(17 25)(18 21)(19 22)(20 23)(26 67)(27 68)(28 69)(29 70)(30 66)(31 156)(32 157)(33 158)(34 159)(35 160)(36 45)(37 41)(38 42)(39 43)(40 44)(46 97)(47 98)(48 99)(49 100)(50 96)(61 108)(62 109)(63 110)(64 106)(65 107)(71 92)(72 93)(73 94)(74 95)(75 91)(76 85)(77 81)(78 82)(79 83)(80 84)(86 101)(87 102)(88 103)(89 104)(90 105)(111 149)(112 150)(113 146)(114 147)(115 148)(116 139)(117 140)(118 136)(119 137)(120 138)(121 134)(122 135)(123 131)(124 132)(125 133)(126 141)(127 142)(128 143)(129 144)(130 145)
(1 81 41 94)(2 82 42 95)(3 83 43 91)(4 84 44 92)(5 85 45 93)(6 111 35 120)(7 112 31 116)(8 113 32 117)(9 114 33 118)(10 115 34 119)(11 63 29 48)(12 64 30 49)(13 65 26 50)(14 61 27 46)(15 62 28 47)(16 105 24 90)(17 101 25 86)(18 102 21 87)(19 103 22 88)(20 104 23 89)(36 72 54 76)(37 73 55 77)(38 74 51 78)(39 75 52 79)(40 71 53 80)(56 107 67 96)(57 108 68 97)(58 109 69 98)(59 110 70 99)(60 106 66 100)(121 144 134 129)(122 145 135 130)(123 141 131 126)(124 142 132 127)(125 143 133 128)(136 154 147 158)(137 155 148 159)(138 151 149 160)(139 152 150 156)(140 153 146 157)
(1 104 30 134)(2 105 26 135)(3 101 27 131)(4 102 28 132)(5 103 29 133)(6 109 151 71)(7 110 152 72)(8 106 153 73)(9 107 154 74)(10 108 155 75)(11 125 45 88)(12 121 41 89)(13 122 42 90)(14 123 43 86)(15 124 44 87)(16 95 145 65)(17 91 141 61)(18 92 142 62)(19 93 143 63)(20 94 144 64)(21 84 127 47)(22 85 128 48)(23 81 129 49)(24 82 130 50)(25 83 126 46)(31 99 156 76)(32 100 157 77)(33 96 158 78)(34 97 159 79)(35 98 160 80)(36 139 59 116)(37 140 60 117)(38 136 56 118)(39 137 57 119)(40 138 58 120)(51 147 67 114)(52 148 68 115)(53 149 69 111)(54 150 70 112)(55 146 66 113)

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

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

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

95 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 5A 5B 5C 5D 8A 8B 8C 8D 10A ··· 10L 10M ··· 10T 20A ··· 20P 20Q ··· 20AB 20AC ··· 20AJ 40A ··· 40P order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 5 5 5 5 8 8 8 8 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 4 4 2 2 2 2 4 4 4 8 8 1 1 1 1 4 4 4 4 1 ··· 1 4 ··· 4 2 ··· 2 4 ··· 4 8 ··· 8 4 ··· 4

95 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 C10 D4 Q8 C4○D4 C4○D8 C5×D4 C5×Q8 C5×C4○D4 C5×C4○D8 C8⋊C22 C5×C8⋊C22 kernel C5×D4.Q8 C5×D4⋊C4 C5×C4⋊C8 C5×C4.Q8 C5×C2.D8 D4×C20 C5×C42.C2 D4.Q8 D4⋊C4 C4⋊C8 C4.Q8 C2.D8 C4×D4 C42.C2 C2×C20 C5×D4 C20 C10 C2×C4 D4 C4 C2 C10 C2 # reps 1 2 1 1 1 1 1 4 8 4 4 4 4 4 2 2 2 4 8 8 8 16 1 4

Matrix representation of C5×D4.Q8 in GL4(𝔽41) generated by

 37 0 0 0 0 37 0 0 0 0 16 0 0 0 0 16
,
 0 1 0 0 40 0 0 0 0 0 40 0 0 0 0 40
,
 0 1 0 0 1 0 0 0 0 0 1 40 0 0 0 40
,
 9 0 0 0 0 9 0 0 0 0 32 9 0 0 0 9
,
 12 29 0 0 29 29 0 0 0 0 1 40 0 0 2 40
G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,16,0,0,0,0,16],[0,40,0,0,1,0,0,0,0,0,40,0,0,0,0,40],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,40,40],[9,0,0,0,0,9,0,0,0,0,32,0,0,0,9,9],[12,29,0,0,29,29,0,0,0,0,1,2,0,0,40,40] >;

C5×D4.Q8 in GAP, Magma, Sage, TeX

C_5\times D_4.Q_8
% in TeX

G:=Group("C5xD4.Q8");
// GroupNames label

G:=SmallGroup(320,979);
// by ID

G=gap.SmallGroup(320,979);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,280,589,1408,1766,10085,2539,124]);
// Polycyclic

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

׿
×
𝔽