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## G = Q8×C5⋊D4order 320 = 26·5

### Direct product of Q8 and C5⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — Q8×C5⋊D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — C2×Q8×D5 — Q8×C5⋊D4
 Lower central C5 — C2×C10 — Q8×C5⋊D4
 Upper central C1 — C22 — C22×Q8

Generators and relations for Q8×C5⋊D4
G = < a,b,c,d,e | a4=c5=d4=e2=1, b2=a2, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 822 in 280 conjugacy classes, 123 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C2×Q8, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C4×D4, C4×Q8, C22⋊Q8, C4⋊Q8, C22×Q8, C22×Q8, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×Q8, C5×Q8, C22×D5, C22×C10, D4×Q8, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C2×Dic10, C2×C4×D5, Q8×D5, C2×C5⋊D4, C22×C20, Q8×C10, Q8×C10, Q8×C10, C20.48D4, C4×C5⋊D4, Dic5⋊Q8, Q8×Dic5, D103Q8, C2×Q8×D5, Q8×C2×C10, Q8×C5⋊D4
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C24, D10, C22×D4, C22×Q8, 2- 1+4, C5⋊D4, C22×D5, D4×Q8, Q8×D5, C2×C5⋊D4, C23×D5, C2×Q8×D5, Q8.10D10, C22×C5⋊D4, Q8×C5⋊D4

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

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

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

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

65 conjugacy classes

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

65 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + + + - + + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 Q8 D4 D5 D10 D10 C5⋊D4 2- 1+4 Q8×D5 Q8.10D10 kernel Q8×C5⋊D4 C20.48D4 C4×C5⋊D4 Dic5⋊Q8 Q8×Dic5 D10⋊3Q8 C2×Q8×D5 Q8×C2×C10 C5⋊D4 C5×Q8 C22×Q8 C22×C4 C2×Q8 Q8 C10 C22 C2 # reps 1 3 3 3 1 3 1 1 4 4 2 6 8 16 1 4 4

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

 40 0 0 0 0 40 0 0 0 0 1 9 0 0 18 40
,
 40 0 0 0 0 40 0 0 0 0 7 19 0 0 6 34
,
 40 40 0 0 8 7 0 0 0 0 1 0 0 0 0 1
,
 3 21 0 0 21 38 0 0 0 0 1 0 0 0 0 1
,
 34 35 0 0 8 7 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,18,0,0,9,40],[40,0,0,0,0,40,0,0,0,0,7,6,0,0,19,34],[40,8,0,0,40,7,0,0,0,0,1,0,0,0,0,1],[3,21,0,0,21,38,0,0,0,0,1,0,0,0,0,1],[34,8,0,0,35,7,0,0,0,0,1,0,0,0,0,1] >;

Q8×C5⋊D4 in GAP, Magma, Sage, TeX

Q_8\times C_5\rtimes D_4
% in TeX

G:=Group("Q8xC5:D4");
// GroupNames label

G:=SmallGroup(320,1487);
// by ID

G=gap.SmallGroup(320,1487);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,387,184,675,12550]);
// Polycyclic

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

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