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## G = C4⋊D4.D5order 320 = 26·5

### 3rd non-split extension by C4⋊D4 of D5 acting via D5/C5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C4⋊D4.D5
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4⋊Dic5 — C2×C4⋊Dic5 — C4⋊D4.D5
 Lower central C5 — C10 — C2×C20 — C4⋊D4.D5
 Upper central C1 — C22 — C22×C4 — C4⋊D4

Generators and relations for C4⋊D4.D5
G = < a,b,c,d,e | a4=b4=c2=d5=1, e2=a2b2, bab-1=cac=eae-1=a-1, ad=da, cbc=b-1, bd=db, ebe-1=a-1b, cd=dc, ece-1=a-1c, ede-1=d-1 >

Subgroups: 382 in 114 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C23, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, C4.Q8, C2×C4⋊C4, C4⋊D4, C52C8, C2×Dic5, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, C23.46D4, C2×C52C8, C4⋊Dic5, C4⋊Dic5, C5×C22⋊C4, C5×C4⋊C4, C22×Dic5, C22×C20, D4×C10, D4×C10, C20.Q8, C20.55D4, D4⋊Dic5, C2×C4⋊Dic5, C5×C4⋊D4, C4⋊D4.D5
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, C4○D4, D10, C22.D4, C2×SD16, C8⋊C22, C5⋊D4, C22×D5, C23.46D4, D4.D5, D42D5, C2×C5⋊D4, C2×D4.D5, C23.18D10, D4⋊D10, C4⋊D4.D5

Smallest permutation representation of C4⋊D4.D5
On 160 points
Generators in S160
```(1 29 9 24)(2 30 10 25)(3 26 6 21)(4 27 7 22)(5 28 8 23)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 66 46 61)(42 67 47 62)(43 68 48 63)(44 69 49 64)(45 70 50 65)(51 76 56 71)(52 77 57 72)(53 78 58 73)(54 79 59 74)(55 80 60 75)(81 101 86 106)(82 102 87 107)(83 103 88 108)(84 104 89 109)(85 105 90 110)(91 111 96 116)(92 112 97 117)(93 113 98 118)(94 114 99 119)(95 115 100 120)(121 141 126 146)(122 142 127 147)(123 143 128 148)(124 144 129 149)(125 145 130 150)(131 151 136 156)(132 152 137 157)(133 153 138 158)(134 154 139 159)(135 155 140 160)
(1 44 14 54)(2 45 15 55)(3 41 11 51)(4 42 12 52)(5 43 13 53)(6 46 16 56)(7 47 17 57)(8 48 18 58)(9 49 19 59)(10 50 20 60)(21 66 31 76)(22 67 32 77)(23 68 33 78)(24 69 34 79)(25 70 35 80)(26 61 36 71)(27 62 37 72)(28 63 38 73)(29 64 39 74)(30 65 40 75)(81 146 91 156)(82 147 92 157)(83 148 93 158)(84 149 94 159)(85 150 95 160)(86 141 96 151)(87 142 97 152)(88 143 98 153)(89 144 99 154)(90 145 100 155)(101 126 111 136)(102 127 112 137)(103 128 113 138)(104 129 114 139)(105 130 115 140)(106 121 116 131)(107 122 117 132)(108 123 118 133)(109 124 119 134)(110 125 120 135)
(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 76)(62 77)(63 78)(64 79)(65 80)(66 71)(67 72)(68 73)(69 74)(70 75)(81 106)(82 107)(83 108)(84 109)(85 110)(86 101)(87 102)(88 103)(89 104)(90 105)(91 116)(92 117)(93 118)(94 119)(95 120)(96 111)(97 112)(98 113)(99 114)(100 115)(121 156)(122 157)(123 158)(124 159)(125 160)(126 151)(127 152)(128 153)(129 154)(130 155)(131 146)(132 147)(133 148)(134 149)(135 150)(136 141)(137 142)(138 143)(139 144)(140 145)
(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)
(1 100 19 85)(2 99 20 84)(3 98 16 83)(4 97 17 82)(5 96 18 81)(6 93 11 88)(7 92 12 87)(8 91 13 86)(9 95 14 90)(10 94 15 89)(21 118 36 103)(22 117 37 102)(23 116 38 101)(24 120 39 105)(25 119 40 104)(26 113 31 108)(27 112 32 107)(28 111 33 106)(29 115 34 110)(30 114 35 109)(41 138 56 123)(42 137 57 122)(43 136 58 121)(44 140 59 125)(45 139 60 124)(46 133 51 128)(47 132 52 127)(48 131 53 126)(49 135 54 130)(50 134 55 129)(61 158 76 143)(62 157 77 142)(63 156 78 141)(64 160 79 145)(65 159 80 144)(66 153 71 148)(67 152 72 147)(68 151 73 146)(69 155 74 150)(70 154 75 149)```

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

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

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

47 conjugacy classes

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

47 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + - - + image C1 C2 C2 C2 C2 C2 D4 D4 D5 C4○D4 SD16 D10 D10 D10 C5⋊D4 C5⋊D4 C8⋊C22 D4⋊2D5 D4.D5 D4⋊D10 kernel C4⋊D4.D5 C20.Q8 C20.55D4 D4⋊Dic5 C2×C4⋊Dic5 C5×C4⋊D4 C2×C20 C22×C10 C4⋊D4 C20 C2×C10 C4⋊C4 C22×C4 C2×D4 C2×C4 C23 C10 C4 C22 C2 # reps 1 2 1 2 1 1 1 1 2 4 4 2 2 2 4 4 1 4 4 4

Matrix representation of C4⋊D4.D5 in GL6(𝔽41)

 0 1 0 0 0 0 40 0 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
,
 40 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 15 4 0 0 0 0 5 26
,
 1 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 13 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 10 0 0 0 0 0 0 37 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 26 15 0 0 0 0 15 15 0 0 0 0 0 0 0 37 0 0 0 0 10 0 0 0 0 0 0 0 29 5 0 0 0 0 12 12

`G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,15,5,0,0,0,0,4,26],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,13,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,37,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[26,15,0,0,0,0,15,15,0,0,0,0,0,0,0,10,0,0,0,0,37,0,0,0,0,0,0,0,29,12,0,0,0,0,5,12] >;`

C4⋊D4.D5 in GAP, Magma, Sage, TeX

`C_4\rtimes D_4.D_5`
`% in TeX`

`G:=Group("C4:D4.D5");`
`// GroupNames label`

`G:=SmallGroup(320,661);`
`// by ID`

`G=gap.SmallGroup(320,661);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,336,254,219,1123,297,136,12550]);`
`// Polycyclic`

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

׿
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