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## G = C10.422- 1+4order 320 = 26·5

### 42nd non-split extension by C10 of 2- 1+4 acting via 2- 1+4/C2×Q8=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C10.422- 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C4×Dic5 — Q8×Dic5 — C10.422- 1+4
 Lower central C5 — C10 — C10.422- 1+4
 Upper central C1 — C22 — C22×Q8

Generators and relations for C10.422- 1+4
G = < a,b,c,d,e | a10=b4=1, c2=a5, d2=a5b2, e2=b2, ab=ba, cac-1=dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 558 in 266 conjugacy classes, 191 normal (11 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×12], C4 [×8], C22, C22 [×2], C22 [×2], C5, C2×C4 [×18], C2×C4 [×8], Q8 [×16], C23, C10, C10 [×2], C10 [×2], C42 [×12], C22⋊C4 [×4], C4⋊C4 [×12], C22×C4 [×3], C2×Q8 [×12], Dic5 [×8], C20 [×12], C2×C10, C2×C10 [×2], C2×C10 [×2], C42⋊C2 [×6], C4×Q8 [×8], C22×Q8, C2×Dic5 [×8], C2×C20 [×18], C5×Q8 [×16], C22×C10, C23.32C23, C4×Dic5 [×12], C4⋊Dic5 [×12], C23.D5 [×4], C22×C20 [×3], Q8×C10 [×12], C23.21D10 [×6], Q8×Dic5 [×8], Q8×C2×C10, C10.422- 1+4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C24, Dic5 [×8], D10 [×7], C23×C4, 2- 1+4 [×2], C2×Dic5 [×28], C22×D5 [×7], C23.32C23, C22×Dic5 [×14], C23×D5, Q8.10D10 [×2], C23×Dic5, C10.422- 1+4

Smallest permutation representation of C10.422- 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 65 30 59)(2 66 21 60)(3 67 22 51)(4 68 23 52)(5 69 24 53)(6 70 25 54)(7 61 26 55)(8 62 27 56)(9 63 28 57)(10 64 29 58)(11 112 157 121)(12 113 158 122)(13 114 159 123)(14 115 160 124)(15 116 151 125)(16 117 152 126)(17 118 153 127)(18 119 154 128)(19 120 155 129)(20 111 156 130)(31 83 47 76)(32 84 48 77)(33 85 49 78)(34 86 50 79)(35 87 41 80)(36 88 42 71)(37 89 43 72)(38 90 44 73)(39 81 45 74)(40 82 46 75)(91 136 110 145)(92 137 101 146)(93 138 102 147)(94 139 103 148)(95 140 104 149)(96 131 105 150)(97 132 106 141)(98 133 107 142)(99 134 108 143)(100 135 109 144)
(1 150 6 145)(2 149 7 144)(3 148 8 143)(4 147 9 142)(5 146 10 141)(11 33 16 38)(12 32 17 37)(13 31 18 36)(14 40 19 35)(15 39 20 34)(21 140 26 135)(22 139 27 134)(23 138 28 133)(24 137 29 132)(25 136 30 131)(41 160 46 155)(42 159 47 154)(43 158 48 153)(44 157 49 152)(45 156 50 151)(51 94 56 99)(52 93 57 98)(53 92 58 97)(54 91 59 96)(55 100 60 95)(61 109 66 104)(62 108 67 103)(63 107 68 102)(64 106 69 101)(65 105 70 110)(71 114 76 119)(72 113 77 118)(73 112 78 117)(74 111 79 116)(75 120 80 115)(81 130 86 125)(82 129 87 124)(83 128 88 123)(84 127 89 122)(85 126 90 121)
(1 111 25 125)(2 120 26 124)(3 119 27 123)(4 118 28 122)(5 117 29 121)(6 116 30 130)(7 115 21 129)(8 114 22 128)(9 113 23 127)(10 112 24 126)(11 69 152 58)(12 68 153 57)(13 67 154 56)(14 66 155 55)(15 65 156 54)(16 64 157 53)(17 63 158 52)(18 62 159 51)(19 61 160 60)(20 70 151 59)(31 103 42 99)(32 102 43 98)(33 101 44 97)(34 110 45 96)(35 109 46 95)(36 108 47 94)(37 107 48 93)(38 106 49 92)(39 105 50 91)(40 104 41 100)(71 134 83 148)(72 133 84 147)(73 132 85 146)(74 131 86 145)(75 140 87 144)(76 139 88 143)(77 138 89 142)(78 137 90 141)(79 136 81 150)(80 135 82 149)
(1 70 30 54)(2 61 21 55)(3 62 22 56)(4 63 23 57)(5 64 24 58)(6 65 25 59)(7 66 26 60)(8 67 27 51)(9 68 28 52)(10 69 29 53)(11 126 157 117)(12 127 158 118)(13 128 159 119)(14 129 160 120)(15 130 151 111)(16 121 152 112)(17 122 153 113)(18 123 154 114)(19 124 155 115)(20 125 156 116)(31 88 47 71)(32 89 48 72)(33 90 49 73)(34 81 50 74)(35 82 41 75)(36 83 42 76)(37 84 43 77)(38 85 44 78)(39 86 45 79)(40 87 46 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)```

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

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

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

74 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4L 4M ··· 4AB 5A 5B 10A ··· 10N 20A ··· 20X order 1 2 2 2 2 2 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 2 ··· 2 10 ··· 10 2 2 2 ··· 2 4 ··· 4

74 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + - + - image C1 C2 C2 C2 C4 D5 D10 Dic5 D10 2- 1+4 Q8.10D10 kernel C10.422- 1+4 C23.21D10 Q8×Dic5 Q8×C2×C10 Q8×C10 C22×Q8 C22×C4 C2×Q8 C2×Q8 C10 C2 # reps 1 6 8 1 16 2 6 16 8 2 8

Matrix representation of C10.422- 1+4 in GL6(𝔽41)

 23 0 0 0 0 0 17 25 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 18 0 0 0 3 40 0 18
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 21 3 0 0 0 0 3 20 0 0 0 0 9 38 23 6 0 0 38 17 21 18
,
 36 38 0 0 0 0 36 5 0 0 0 0 0 0 9 38 23 6 0 0 19 21 17 40 0 0 21 3 0 0 0 0 25 24 15 11
,
 5 3 0 0 0 0 5 36 0 0 0 0 0 0 3 40 1 2 0 0 0 0 1 0 0 0 0 40 0 0 0 0 36 2 40 38
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 21 3 0 0 0 0 3 20 0 0 0 0 32 3 18 35 0 0 25 35 20 23

`G:=sub<GL(6,GF(41))| [23,17,0,0,0,0,0,25,0,0,0,0,0,0,16,0,0,3,0,0,0,16,0,40,0,0,0,0,18,0,0,0,0,0,0,18],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,21,3,9,38,0,0,3,20,38,17,0,0,0,0,23,21,0,0,0,0,6,18],[36,36,0,0,0,0,38,5,0,0,0,0,0,0,9,19,21,25,0,0,38,21,3,24,0,0,23,17,0,15,0,0,6,40,0,11],[5,5,0,0,0,0,3,36,0,0,0,0,0,0,3,0,0,36,0,0,40,0,40,2,0,0,1,1,0,40,0,0,2,0,0,38],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,21,3,32,25,0,0,3,20,3,35,0,0,0,0,18,20,0,0,0,0,35,23] >;`

C10.422- 1+4 in GAP, Magma, Sage, TeX

`C_{10}._{42}2_-^{1+4}`
`% in TeX`

`G:=Group("C10.42ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1484);`
`// by ID`

`G=gap.SmallGroup(320,1484);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,232,387,184,1123,12550]);`
`// Polycyclic`

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

׿
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