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## G = (C5×D4)⋊14D4order 320 = 26·5

### 2nd semidirect product of C5×D4 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — (C5×D4)⋊14D4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×D20 — C2×D4⋊D5 — (C5×D4)⋊14D4
 Lower central C5 — C10 — C2×C20 — (C5×D4)⋊14D4
 Upper central C1 — C22 — C22×C4 — C2×C4○D4

Generators and relations for (C5×D4)⋊14D4
G = < a,b,c,d,e | a5=b4=c2=d4=e2=1, ab=ba, ac=ca, dad-1=eae=a-1, cbc=dbd-1=ebe=b-1, dcd-1=b-1c, ece=bc, ede=d-1 >

Subgroups: 590 in 162 conjugacy classes, 47 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, C52C8, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×D5, C22×C10, C22×C10, D4⋊D4, C2×C52C8, C4⋊Dic5, D10⋊C4, D4⋊D5, Q8⋊D5, C2×D20, C2×C5⋊D4, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4, C20.55D4, D4⋊Dic5, Q8⋊Dic5, C207D4, C2×D4⋊D5, C2×Q8⋊D5, C10×C4○D4, (C5×D4)⋊14D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, C4○D8, C8⋊C22, C5⋊D4, C22×D5, D4⋊D4, C2×C5⋊D4, D4⋊D10, D4.8D10, C242D5, (C5×D4)⋊14D4

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

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

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

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

59 conjugacy classes

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

59 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 D5 D10 D10 D10 C4○D8 C5⋊D4 C5⋊D4 C5⋊D4 C5⋊D4 C8⋊C22 D4⋊D10 D4.8D10 kernel (C5×D4)⋊14D4 C20.55D4 D4⋊Dic5 Q8⋊Dic5 C20⋊7D4 C2×D4⋊D5 C2×Q8⋊D5 C10×C4○D4 C2×C20 C5×D4 C5×Q8 C22×C10 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C10 C2×C4 D4 Q8 C23 C10 C2 C2 # reps 1 1 1 1 1 1 1 1 1 2 2 1 2 2 2 2 4 4 8 8 4 1 4 4

Matrix representation of (C5×D4)⋊14D4 in GL4(𝔽41) generated by

 6 40 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 1 4 0 0 20 40
,
 1 0 0 0 0 1 0 0 0 0 17 34 0 0 6 24
,
 18 35 0 0 20 23 0 0 0 0 9 0 0 0 16 32
,
 1 0 0 0 6 40 0 0 0 0 1 4 0 0 0 40
G:=sub<GL(4,GF(41))| [6,1,0,0,40,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,20,0,0,4,40],[1,0,0,0,0,1,0,0,0,0,17,6,0,0,34,24],[18,20,0,0,35,23,0,0,0,0,9,16,0,0,0,32],[1,6,0,0,0,40,0,0,0,0,1,0,0,0,4,40] >;

(C5×D4)⋊14D4 in GAP, Magma, Sage, TeX

(C_5\times D_4)\rtimes_{14}D_4
% in TeX

G:=Group("(C5xD4):14D4");
// GroupNames label

G:=SmallGroup(320,865);
// by ID

G=gap.SmallGroup(320,865);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,184,1684,851,102,12550]);
// Polycyclic

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

׿
×
𝔽