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G = C4×Q8×D5order 320 = 26·5

Direct product of C4, Q8 and D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C4×Q8×D5
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — C2×Q8×D5 — C4×Q8×D5
 Lower central C5 — C10 — C4×Q8×D5
 Upper central C1 — C2×C4 — C4×Q8

Generators and relations for C4×Q8×D5
G = < a,b,c,d,e | a4=b4=d5=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 814 in 298 conjugacy classes, 169 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×14], C22, C22 [×6], C5, C2×C4, C2×C4 [×6], C2×C4 [×29], Q8 [×4], Q8 [×12], C23, D5 [×4], C10 [×3], C42 [×3], C42 [×9], C4⋊C4 [×3], C4⋊C4 [×9], C22×C4 [×7], C2×Q8, C2×Q8 [×11], Dic5 [×8], Dic5 [×3], C20 [×8], C20 [×3], D10 [×6], C2×C10, C2×C42 [×3], C2×C4⋊C4 [×3], C4×Q8, C4×Q8 [×7], C22×Q8, Dic10 [×12], C4×D5 [×16], C4×D5 [×6], C2×Dic5, C2×Dic5 [×6], C2×C20, C2×C20 [×6], C5×Q8 [×4], C22×D5, C2×C4×Q8, C4×Dic5 [×9], C10.D4 [×6], C4⋊Dic5 [×3], C4×C20 [×3], C5×C4⋊C4 [×3], C2×Dic10 [×3], C2×C4×D5, C2×C4×D5 [×6], Q8×D5 [×8], Q8×C10, C4×Dic10 [×3], D5×C42 [×3], Dic53Q8 [×3], D5×C4⋊C4 [×3], Q8×Dic5, Q8×C20, C2×Q8×D5, C4×Q8×D5
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], Q8 [×4], C23 [×15], D5, C22×C4 [×14], C2×Q8 [×6], C4○D4 [×2], C24, D10 [×7], C4×Q8 [×4], C23×C4, C22×Q8, C2×C4○D4, C4×D5 [×4], C22×D5 [×7], C2×C4×Q8, C2×C4×D5 [×6], Q8×D5 [×2], C23×D5, D5×C22×C4, C2×Q8×D5, D5×C4○D4, C4×Q8×D5

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

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4P 4Q 4R 4S 4T 4U ··· 4AF 5A 5B 10A ··· 10F 20A ··· 20H 20I ··· 20AF order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 4 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 5 5 5 5 1 1 1 1 2 ··· 2 5 5 5 5 10 ··· 10 2 2 2 ··· 2 2 ··· 2 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + - + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C4 Q8 D5 C4○D4 D10 D10 D10 C4×D5 Q8×D5 D5×C4○D4 kernel C4×Q8×D5 C4×Dic10 D5×C42 Dic5⋊3Q8 D5×C4⋊C4 Q8×Dic5 Q8×C20 C2×Q8×D5 Q8×D5 C4×D5 C4×Q8 D10 C42 C4⋊C4 C2×Q8 Q8 C4 C2 # reps 1 3 3 3 3 1 1 1 16 4 2 4 6 6 2 16 4 4

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

 9 0 0 0 0 9 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 13 5 0 0 7 28
,
 1 0 0 0 0 1 0 0 0 0 39 30 0 0 34 2
,
 35 1 0 0 5 40 0 0 0 0 1 0 0 0 0 1
,
 40 40 0 0 0 1 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,13,7,0,0,5,28],[1,0,0,0,0,1,0,0,0,0,39,34,0,0,30,2],[35,5,0,0,1,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,40,1,0,0,0,0,1,0,0,0,0,1] >;

C4×Q8×D5 in GAP, Magma, Sage, TeX

C_4\times Q_8\times D_5
% in TeX

G:=Group("C4xQ8xD5");
// GroupNames label

G:=SmallGroup(320,1243);
// by ID

G=gap.SmallGroup(320,1243);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,184,80,12550]);
// Polycyclic

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

׿
×
𝔽