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G = C22⋊C4×Dic5order 320 = 26·5

Direct product of C22⋊C4 and Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C22⋊C4×Dic5
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C22×Dic5 — C23×Dic5 — C22⋊C4×Dic5
 Lower central C5 — C10 — C22⋊C4×Dic5
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C22⋊C4×Dic5
G = < a,b,c,d,e | a2=b2=c4=d10=1, e2=d5, cac-1=ab=ba, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 734 in 258 conjugacy classes, 111 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×14], C22 [×3], C22 [×8], C22 [×12], C5, C2×C4 [×4], C2×C4 [×30], C23, C23 [×6], C23 [×4], C10 [×3], C10 [×4], C10 [×4], C42 [×4], C22⋊C4 [×4], C22⋊C4 [×4], C22×C4 [×2], C22×C4 [×12], C24, Dic5 [×4], Dic5 [×6], C20 [×4], C2×C10 [×3], C2×C10 [×8], C2×C10 [×12], C2.C42 [×2], C2×C42 [×2], C2×C22⋊C4, C2×C22⋊C4, C23×C4, C2×Dic5 [×12], C2×Dic5 [×14], C2×C20 [×4], C2×C20 [×4], C22×C10, C22×C10 [×6], C22×C10 [×4], C4×C22⋊C4, C4×Dic5 [×4], C23.D5 [×4], C5×C22⋊C4 [×4], C22×Dic5 [×2], C22×Dic5 [×6], C22×Dic5 [×4], C22×C20 [×2], C23×C10, C10.10C42 [×2], C2×C4×Dic5 [×2], C2×C23.D5, C10×C22⋊C4, C23×Dic5, C22⋊C4×Dic5
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×4], C23, D5, C42 [×4], C22⋊C4 [×4], C22×C4 [×3], C2×D4 [×2], C4○D4 [×2], Dic5 [×4], D10 [×3], C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4 [×4], C4×D5 [×4], C2×Dic5 [×6], C22×D5, C4×C22⋊C4, C4×Dic5 [×4], C2×C4×D5 [×2], D4×D5 [×2], D42D5 [×2], C22×Dic5, C23.11D10, D5×C22⋊C4, Dic54D4 [×2], C2×C4×Dic5, D4×Dic5 [×2], C22⋊C4×Dic5

Smallest permutation representation of C22⋊C4×Dic5
On 160 points
Generators in S160
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 11)(10 12)(21 137)(22 138)(23 139)(24 140)(25 131)(26 132)(27 133)(28 134)(29 135)(30 136)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 41)(38 42)(39 43)(40 44)(51 83)(52 84)(53 85)(54 86)(55 87)(56 88)(57 89)(58 90)(59 81)(60 82)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 101)(98 102)(99 103)(100 104)(111 122)(112 123)(113 124)(114 125)(115 126)(116 127)(117 128)(118 129)(119 130)(120 121)(141 151)(142 152)(143 153)(144 154)(145 155)(146 156)(147 157)(148 158)(149 159)(150 160)
(1 39)(2 40)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 141)(22 142)(23 143)(24 144)(25 145)(26 146)(27 147)(28 148)(29 149)(30 150)(51 72)(52 73)(53 74)(54 75)(55 76)(56 77)(57 78)(58 79)(59 80)(60 71)(61 82)(62 83)(63 84)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 81)(91 115)(92 116)(93 117)(94 118)(95 119)(96 120)(97 111)(98 112)(99 113)(100 114)(101 122)(102 123)(103 124)(104 125)(105 126)(106 127)(107 128)(108 129)(109 130)(110 121)(131 155)(132 156)(133 157)(134 158)(135 159)(136 160)(137 151)(138 152)(139 153)(140 154)
(1 64 18 58)(2 65 19 59)(3 66 20 60)(4 67 11 51)(5 68 12 52)(6 69 13 53)(7 70 14 54)(8 61 15 55)(9 62 16 56)(10 63 17 57)(21 122 156 116)(22 123 157 117)(23 124 158 118)(24 125 159 119)(25 126 160 120)(26 127 151 111)(27 128 152 112)(28 129 153 113)(29 130 154 114)(30 121 155 115)(31 87 50 71)(32 88 41 72)(33 89 42 73)(34 90 43 74)(35 81 44 75)(36 82 45 76)(37 83 46 77)(38 84 47 78)(39 85 48 79)(40 86 49 80)(91 150 110 131)(92 141 101 132)(93 142 102 133)(94 143 103 134)(95 144 104 135)(96 145 105 136)(97 146 106 137)(98 147 107 138)(99 148 108 139)(100 149 109 140)
(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 92 6 97)(2 91 7 96)(3 100 8 95)(4 99 9 94)(5 98 10 93)(11 108 16 103)(12 107 17 102)(13 106 18 101)(14 105 19 110)(15 104 20 109)(21 90 26 85)(22 89 27 84)(23 88 28 83)(24 87 29 82)(25 86 30 81)(31 114 36 119)(32 113 37 118)(33 112 38 117)(34 111 39 116)(35 120 40 115)(41 129 46 124)(42 128 47 123)(43 127 48 122)(44 126 49 121)(45 125 50 130)(51 139 56 134)(52 138 57 133)(53 137 58 132)(54 136 59 131)(55 135 60 140)(61 144 66 149)(62 143 67 148)(63 142 68 147)(64 141 69 146)(65 150 70 145)(71 154 76 159)(72 153 77 158)(73 152 78 157)(74 151 79 156)(75 160 80 155)

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

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

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

80 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I ··· 4P 4Q ··· 4AB 5A 5B 10A ··· 10N 10O ··· 10V 20A ··· 20P order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 4 ··· 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 2 2 2 2 ··· 2 5 ··· 5 10 ··· 10 2 2 2 ··· 2 4 ··· 4 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 C4 C4 C4 D4 D5 C4○D4 Dic5 D10 D10 C4×D5 D4×D5 D4⋊2D5 kernel C22⋊C4×Dic5 C10.10C42 C2×C4×Dic5 C2×C23.D5 C10×C22⋊C4 C23×Dic5 C23.D5 C5×C22⋊C4 C22×Dic5 C2×Dic5 C2×C22⋊C4 C2×C10 C22⋊C4 C22×C4 C24 C23 C22 C22 # reps 1 2 2 1 1 1 8 8 8 4 2 4 8 4 2 16 4 4

Matrix representation of C22⋊C4×Dic5 in GL5(𝔽41)

 40 0 0 0 0 0 1 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 40
,
 1 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 1 0 0 0 0 0 1
,
 9 0 0 0 0 0 0 1 0 0 0 40 0 0 0 0 0 0 9 0 0 0 0 0 9
,
 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 0 0 0 35 36 0 0 0 40 40
,
 32 0 0 0 0 0 32 0 0 0 0 0 32 0 0 0 0 0 39 37 0 0 0 11 2

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[9,0,0,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,9],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,35,40,0,0,0,36,40],[32,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,39,11,0,0,0,37,2] >;

C22⋊C4×Dic5 in GAP, Magma, Sage, TeX

C_2^2\rtimes C_4\times {\rm Dic}_5
% in TeX

G:=Group("C2^2:C4xDic5");
// GroupNames label

G:=SmallGroup(320,568);
// by ID

G=gap.SmallGroup(320,568);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,387,100,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^10=1,e^2=d^5,c*a*c^-1=a*b=b*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

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