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## G = C10.1042- 1+4order 320 = 26·5

### 59th non-split extension by C10 of 2- 1+4 acting via 2- 1+4/C4○D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C10.1042- 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C2×C10.D4 — C10.1042- 1+4
 Lower central C5 — C2×C10 — C10.1042- 1+4
 Upper central C1 — C22 — C2×C4○D4

Generators and relations for C10.1042- 1+4
G = < a,b,c,d,e | a10=b4=1, c2=a5, d2=e2=b2, bab-1=cac-1=eae-1=a-1, ad=da, cbc-1=a5b-1, bd=db, be=eb, cd=dc, ece-1=a5c, ede-1=b2d >

Subgroups: 902 in 292 conjugacy classes, 113 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22.D4, C4⋊Q8, C2×C4○D4, C2×C4○D4, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×D5, C22×C10, C22×C10, D46D4, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, D10⋊C4, D10⋊C4, C23.D5, C23.D5, C2×Dic10, C2×C4×D5, D42D5, C22×Dic5, C2×C5⋊D4, C2×C5⋊D4, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4, C2×C10.D4, C20.48D4, C4×C5⋊D4, C23.23D10, D4×Dic5, C23.18D10, Dic5⋊D4, Dic5⋊Q8, D103Q8, C2×D42D5, C10×C4○D4, C10.1042- 1+4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C22×D4, C2×C4○D4, 2- 1+4, C5⋊D4, C22×D5, D46D4, C2×C5⋊D4, C23×D5, D5×C4○D4, D4.10D10, C22×C5⋊D4, C10.1042- 1+4

Smallest permutation representation of C10.1042- 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)
(1 130 28 118)(2 129 29 117)(3 128 30 116)(4 127 21 115)(5 126 22 114)(6 125 23 113)(7 124 24 112)(8 123 25 111)(9 122 26 120)(10 121 27 119)(11 65 153 53)(12 64 154 52)(13 63 155 51)(14 62 156 60)(15 61 157 59)(16 70 158 58)(17 69 159 57)(18 68 160 56)(19 67 151 55)(20 66 152 54)(31 102 43 100)(32 101 44 99)(33 110 45 98)(34 109 46 97)(35 108 47 96)(36 107 48 95)(37 106 49 94)(38 105 50 93)(39 104 41 92)(40 103 42 91)(71 135 83 147)(72 134 84 146)(73 133 85 145)(74 132 86 144)(75 131 87 143)(76 140 88 142)(77 139 89 141)(78 138 90 150)(79 137 81 149)(80 136 82 148)
(1 65 6 70)(2 64 7 69)(3 63 8 68)(4 62 9 67)(5 61 10 66)(11 130 16 125)(12 129 17 124)(13 128 18 123)(14 127 19 122)(15 126 20 121)(21 60 26 55)(22 59 27 54)(23 58 28 53)(24 57 29 52)(25 56 30 51)(31 87 36 82)(32 86 37 81)(33 85 38 90)(34 84 39 89)(35 83 40 88)(41 77 46 72)(42 76 47 71)(43 75 48 80)(44 74 49 79)(45 73 50 78)(91 147 96 142)(92 146 97 141)(93 145 98 150)(94 144 99 149)(95 143 100 148)(101 137 106 132)(102 136 107 131)(103 135 108 140)(104 134 109 139)(105 133 110 138)(111 155 116 160)(112 154 117 159)(113 153 118 158)(114 152 119 157)(115 151 120 156)
(1 130 28 118)(2 121 29 119)(3 122 30 120)(4 123 21 111)(5 124 22 112)(6 125 23 113)(7 126 24 114)(8 127 25 115)(9 128 26 116)(10 129 27 117)(11 58 153 70)(12 59 154 61)(13 60 155 62)(14 51 156 63)(15 52 157 64)(16 53 158 65)(17 54 159 66)(18 55 160 67)(19 56 151 68)(20 57 152 69)(31 96 43 108)(32 97 44 109)(33 98 45 110)(34 99 46 101)(35 100 47 102)(36 91 48 103)(37 92 49 104)(38 93 50 105)(39 94 41 106)(40 95 42 107)(71 136 83 148)(72 137 84 149)(73 138 85 150)(74 139 86 141)(75 140 87 142)(76 131 88 143)(77 132 89 144)(78 133 90 145)(79 134 81 146)(80 135 82 147)
(1 98 28 110)(2 97 29 109)(3 96 30 108)(4 95 21 107)(5 94 22 106)(6 93 23 105)(7 92 24 104)(8 91 25 103)(9 100 26 102)(10 99 27 101)(11 85 153 73)(12 84 154 72)(13 83 155 71)(14 82 156 80)(15 81 157 79)(16 90 158 78)(17 89 159 77)(18 88 160 76)(19 87 151 75)(20 86 152 74)(31 120 43 122)(32 119 44 121)(33 118 45 130)(34 117 46 129)(35 116 47 128)(36 115 48 127)(37 114 49 126)(38 113 50 125)(39 112 41 124)(40 111 42 123)(51 135 63 147)(52 134 64 146)(53 133 65 145)(54 132 66 144)(55 131 67 143)(56 140 68 142)(57 139 69 141)(58 138 70 150)(59 137 61 149)(60 136 62 148)```

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

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

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

65 conjugacy classes

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

65 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 D10 C5⋊D4 2- 1+4 D5×C4○D4 D4.10D10 kernel C10.1042- 1+4 C2×C10.D4 C20.48D4 C4×C5⋊D4 C23.23D10 D4×Dic5 C23.18D10 Dic5⋊D4 Dic5⋊Q8 D10⋊3Q8 C2×D4⋊2D5 C10×C4○D4 C5×D4 C2×C4○D4 Dic5 C22×C4 C2×D4 C2×Q8 D4 C10 C2 C2 # reps 1 2 1 1 2 1 2 2 1 1 1 1 4 2 4 6 6 2 16 1 4 4

Matrix representation of C10.1042- 1+4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 6 7 0 0 0 0 35 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 40 0 0 0 0 0 1 1 0 0 0 0 0 0 35 34 0 0 0 0 5 6 0 0 0 0 0 0 9 0 0 0 0 0 0 9
,
 1 2 0 0 0 0 40 40 0 0 0 0 0 0 35 34 0 0 0 0 5 6 0 0 0 0 0 0 32 0 0 0 0 0 0 9
,
 40 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 9 0 0 0 0 0 0 32
,
 40 0 0 0 0 0 1 1 0 0 0 0 0 0 6 7 0 0 0 0 36 35 0 0 0 0 0 0 0 9 0 0 0 0 9 0

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,6,35,0,0,0,0,7,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,1,0,0,0,0,0,1,0,0,0,0,0,0,35,5,0,0,0,0,34,6,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,40,0,0,0,0,2,40,0,0,0,0,0,0,35,5,0,0,0,0,34,6,0,0,0,0,0,0,32,0,0,0,0,0,0,9],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,0,32],[40,1,0,0,0,0,0,1,0,0,0,0,0,0,6,36,0,0,0,0,7,35,0,0,0,0,0,0,0,9,0,0,0,0,9,0] >;`

C10.1042- 1+4 in GAP, Magma, Sage, TeX

`C_{10}._{104}2_-^{1+4}`
`% in TeX`

`G:=Group("C10.104ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1496);`
`// by ID`

`G=gap.SmallGroup(320,1496);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,758,100,346,12550]);`
`// Polycyclic`

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

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