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## G = (C5×D4).32D4order 320 = 26·5

### 2nd non-split extension by C5×D4 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — (C5×D4).32D4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×Dic10 — C2×D4.D5 — (C5×D4).32D4
 Lower central C5 — C10 — C2×C20 — (C5×D4).32D4
 Upper central C1 — C22 — C22×C4 — C2×C4○D4

Generators and relations for (C5×D4).32D4
G = < a,b,c,d,e | a5=b4=c2=d4=1, e2=b2, ab=ba, ac=ca, dad-1=eae-1=a-1, cbc=dbd-1=ebe-1=b-1, dcd-1=ece-1=bc, ede-1=d-1 >

Subgroups: 446 in 152 conjugacy classes, 47 normal (39 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×5], C22, C22 [×7], C5, C8 [×2], C2×C4 [×2], C2×C4 [×9], D4 [×2], D4 [×5], Q8 [×2], Q8 [×3], C23, C23, C10 [×3], C10 [×3], C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], SD16 [×2], Q16 [×2], C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4 [×4], Dic5 [×2], C20 [×2], C20 [×3], C2×C10, C2×C10 [×7], C22⋊C8, D4⋊C4, Q8⋊C4, C22⋊Q8, C2×SD16, C2×Q16, C2×C4○D4, C52C8 [×2], Dic10 [×2], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×7], C5×D4 [×2], C5×D4 [×5], C5×Q8 [×2], C5×Q8, C22×C10, C22×C10, D4.7D4, C2×C52C8 [×2], C10.D4, C4⋊Dic5, D4.D5 [×2], C5⋊Q16 [×2], C23.D5, C2×Dic10, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4 [×4], C20.55D4, D4⋊Dic5, Q8⋊Dic5, C20.48D4, C2×D4.D5, C2×C5⋊Q16, C10×C4○D4, (C5×D4).32D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, C4○D8, C8.C22, C5⋊D4 [×6], C22×D5, D4.7D4, C2×C5⋊D4 [×3], D4.8D10, D4.9D10, C242D5, (C5×D4).32D4

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

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

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

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

59 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 ··· 10R 20A ··· 20H 20I ··· 20T 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 20 ··· 20 20 ··· 20 size 1 1 1 1 4 4 4 2 2 2 2 4 4 40 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 D4.8D10 D4.9D10 kernel (C5×D4).32D4 C20.55D4 D4⋊Dic5 Q8⋊Dic5 C20.48D4 C2×D4.D5 C2×C5⋊Q16 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).32D4 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 16 0 0 0 12 18
,
 9 0 0 0 36 32 0 0 0 0 40 0 0 0 0 40
,
 9 16 0 0 36 32 0 0 0 0 40 0 0 0 12 1
,
 38 22 0 0 22 3 0 0 0 0 21 24 0 0 26 20
,
 14 34 0 0 34 27 0 0 0 0 20 17 0 0 32 21
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,16,12,0,0,0,18],[9,36,0,0,0,32,0,0,0,0,40,0,0,0,0,40],[9,36,0,0,16,32,0,0,0,0,40,12,0,0,0,1],[38,22,0,0,22,3,0,0,0,0,21,26,0,0,24,20],[14,34,0,0,34,27,0,0,0,0,20,32,0,0,17,21] >;

(C5×D4).32D4 in GAP, Magma, Sage, TeX

(C_5\times D_4)._{32}D_4
% in TeX

G:=Group("(C5xD4).32D4");
// GroupNames label

G:=SmallGroup(320,866);
// by ID

G=gap.SmallGroup(320,866);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,254,184,1684,851,102,12550]);
// Polycyclic

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

׿
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