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G = C10.2+ 1+4order 320 = 26·5

9th non-split extension by C10 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C10.42- 1+4, C10.92+ 1+4, (C2×C4)⋊4D20, (C2×C20)⋊9D4, C4⋊D209C2, C207D45C2, C4.69(C2×D20), C4⋊C4.263D10, D102Q88C2, C20.222(C2×D4), C22.7(C2×D20), (C2×C10).54C24, C10.11(C22×D4), C2.13(C22×D20), (C2×C20).137C23, (C22×C4).179D10, C4⋊Dic5.29C22, C2.12(D46D10), C22.88(C23×D5), D10⋊C4.1C22, (C2×D20).262C22, (C22×C20).75C22, (C2×Dic5).16C23, (C22×D5).13C23, C23.226(C22×D5), (C22×C10).403C23, C2.7(Q8.10D10), C51(C22.31C24), (C2×Dic10).290C22, (C2×C4⋊C4)⋊19D5, (C10×C4⋊C4)⋊16C2, (C2×C4○D20)⋊16C2, (C2×C4×D5).64C22, (C2×C10).176(C2×D4), (C5×C4⋊C4).296C22, (C2×C4).572(C22×D5), (C2×C5⋊D4).101C22, SmallGroup(320,1182)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C10.2+ 1+4
C1C5C10C2×C10C22×D5C2×C4×D5C2×C4○D20 — C10.2+ 1+4
C5C2×C10 — C10.2+ 1+4
C1C22C2×C4⋊C4

Generators and relations for C10.2+ 1+4
 G = < a,b,c,d,e | a10=b4=e2=1, c2=a5, d2=b2, bab-1=dad-1=eae=a-1, ac=ca, cbc-1=a5b-1, dbd-1=a5b, be=eb, dcd-1=ece=a5c, ede=a5b2d >

Subgroups: 1182 in 294 conjugacy classes, 111 normal (17 characteristic)
C1, C2 [×3], C2 [×6], C4 [×4], C4 [×8], C22, C22 [×2], C22 [×14], C5, C2×C4 [×10], C2×C4 [×14], D4 [×16], Q8 [×4], C23, C23 [×4], D5 [×4], C10 [×3], C10 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4 [×10], C2×Q8 [×2], C4○D4 [×8], Dic5 [×4], C20 [×4], C20 [×4], D10 [×12], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C4⋊C4, C4⋊D4 [×8], C22⋊Q8 [×4], C2×C4○D4 [×2], Dic10 [×4], C4×D5 [×8], D20 [×8], C2×Dic5 [×4], C5⋊D4 [×8], C2×C20 [×10], C2×C20 [×2], C22×D5 [×4], C22×C10, C22.31C24, C4⋊Dic5 [×4], D10⋊C4 [×8], C5×C4⋊C4 [×4], C2×Dic10 [×2], C2×C4×D5 [×4], C2×D20 [×6], C4○D20 [×8], C2×C5⋊D4 [×4], C22×C20, C22×C20 [×2], C4⋊D20 [×4], D102Q8 [×4], C207D4 [×4], C10×C4⋊C4, C2×C4○D20 [×2], C10.2+ 1+4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, 2+ 1+4, 2- 1+4, D20 [×4], C22×D5 [×7], C22.31C24, C2×D20 [×6], C23×D5, C22×D20, D46D10, Q8.10D10, C10.2+ 1+4

Smallest permutation representation of C10.2+ 1+4
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 70 30 56)(2 69 21 55)(3 68 22 54)(4 67 23 53)(5 66 24 52)(6 65 25 51)(7 64 26 60)(8 63 27 59)(9 62 28 58)(10 61 29 57)(11 130 151 116)(12 129 152 115)(13 128 153 114)(14 127 154 113)(15 126 155 112)(16 125 156 111)(17 124 157 120)(18 123 158 119)(19 122 159 118)(20 121 160 117)(31 76 45 90)(32 75 46 89)(33 74 47 88)(34 73 48 87)(35 72 49 86)(36 71 50 85)(37 80 41 84)(38 79 42 83)(39 78 43 82)(40 77 44 81)(91 131 105 145)(92 140 106 144)(93 139 107 143)(94 138 108 142)(95 137 109 141)(96 136 110 150)(97 135 101 149)(98 134 102 148)(99 133 103 147)(100 132 104 146)
(1 31 6 36)(2 32 7 37)(3 33 8 38)(4 34 9 39)(5 35 10 40)(11 131 16 136)(12 132 17 137)(13 133 18 138)(14 134 19 139)(15 135 20 140)(21 46 26 41)(22 47 27 42)(23 48 28 43)(24 49 29 44)(25 50 30 45)(51 76 56 71)(52 77 57 72)(53 78 58 73)(54 79 59 74)(55 80 60 75)(61 86 66 81)(62 87 67 82)(63 88 68 83)(64 89 69 84)(65 90 70 85)(91 130 96 125)(92 121 97 126)(93 122 98 127)(94 123 99 128)(95 124 100 129)(101 112 106 117)(102 113 107 118)(103 114 108 119)(104 115 109 120)(105 116 110 111)(141 152 146 157)(142 153 147 158)(143 154 148 159)(144 155 149 160)(145 156 150 151)
(1 16 30 156)(2 15 21 155)(3 14 22 154)(4 13 23 153)(5 12 24 152)(6 11 25 151)(7 20 26 160)(8 19 27 159)(9 18 28 158)(10 17 29 157)(31 131 45 145)(32 140 46 144)(33 139 47 143)(34 138 48 142)(35 137 49 141)(36 136 50 150)(37 135 41 149)(38 134 42 148)(39 133 43 147)(40 132 44 146)(51 111 65 125)(52 120 66 124)(53 119 67 123)(54 118 68 122)(55 117 69 121)(56 116 70 130)(57 115 61 129)(58 114 62 128)(59 113 63 127)(60 112 64 126)(71 105 85 91)(72 104 86 100)(73 103 87 99)(74 102 88 98)(75 101 89 97)(76 110 90 96)(77 109 81 95)(78 108 82 94)(79 107 83 93)(80 106 84 92)
(1 110)(2 109)(3 108)(4 107)(5 106)(6 105)(7 104)(8 103)(9 102)(10 101)(11 76)(12 75)(13 74)(14 73)(15 72)(16 71)(17 80)(18 79)(19 78)(20 77)(21 95)(22 94)(23 93)(24 92)(25 91)(26 100)(27 99)(28 98)(29 97)(30 96)(31 116)(32 115)(33 114)(34 113)(35 112)(36 111)(37 120)(38 119)(39 118)(40 117)(41 124)(42 123)(43 122)(44 121)(45 130)(46 129)(47 128)(48 127)(49 126)(50 125)(51 131)(52 140)(53 139)(54 138)(55 137)(56 136)(57 135)(58 134)(59 133)(60 132)(61 149)(62 148)(63 147)(64 146)(65 145)(66 144)(67 143)(68 142)(69 141)(70 150)(81 160)(82 159)(83 158)(84 157)(85 156)(86 155)(87 154)(88 153)(89 152)(90 151)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L5A5B10A···10N20A···20X
order12222222224444444444445510···1020···20
size111122202020202222444420202020222···24···4

62 irreducible representations

dim111111222224444
type++++++++++++-
imageC1C2C2C2C2C2D4D5D10D10D202+ 1+42- 1+4D46D10Q8.10D10
kernelC10.2+ 1+4C4⋊D20D102Q8C207D4C10×C4⋊C4C2×C4○D20C2×C20C2×C4⋊C4C4⋊C4C22×C4C2×C4C10C10C2C2
# reps1444124286161144

Matrix representation of C10.2+ 1+4 in GL6(𝔽41)

4000000
0400000
00353400
006000
00003534
000060
,
100000
8400000
00213800
00382000
00002138
00003820
,
20360000
31210000
000010
000001
0040000
0004000
,
20360000
6210000
003241821
0018382623
0018213817
002623233
,
100000
8400000
0022321022
0032192231
001022199
002231922

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,6,0,0,0,0,34,0,0,0,0,0,0,0,35,6,0,0,0,0,34,0],[1,8,0,0,0,0,0,40,0,0,0,0,0,0,21,38,0,0,0,0,38,20,0,0,0,0,0,0,21,38,0,0,0,0,38,20],[20,31,0,0,0,0,36,21,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0],[20,6,0,0,0,0,36,21,0,0,0,0,0,0,3,18,18,26,0,0,24,38,21,23,0,0,18,26,38,23,0,0,21,23,17,3],[1,8,0,0,0,0,0,40,0,0,0,0,0,0,22,32,10,22,0,0,32,19,22,31,0,0,10,22,19,9,0,0,22,31,9,22] >;

C10.2+ 1+4 in GAP, Magma, Sage, TeX

C_{10}.2_+^{1+4}
% in TeX

G:=Group("C10.ES+(2,2)");
// GroupNames label

G:=SmallGroup(320,1182);
// by ID

G=gap.SmallGroup(320,1182);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,675,570,80,12550]);
// Polycyclic

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

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