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## G = C10.502+ 1+4order 320 = 26·5

### 50th non-split extension by C10 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C10.502+ 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C23.11D10 — C10.502+ 1+4
 Lower central C5 — C2×C10 — C10.502+ 1+4
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for C10.502+ 1+4
G = < a,b,c,d,e | a10=b4=e2=1, c2=a5, d2=a5b2, bab-1=a-1, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=a5c, ce=ec, ede=b2d >

Subgroups: 622 in 206 conjugacy classes, 103 normal (31 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, Q8, C23, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2×Q8, Dic5, Dic5, C20, C2×C10, C2×C10, C2×C10, C2×C4⋊C4, C42⋊C2, C22⋊Q8, C22⋊Q8, C42.C2, C4⋊Q8, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C2×C20, C5×Q8, C22×C10, C23.41C23, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×Dic10, C22×Dic5, C22×C20, Q8×C10, C23.11D10, Dic5.14D4, C20⋊Q8, Dic5.Q8, C2×C10.D4, C20.48D4, Dic5⋊Q8, C5×C22⋊Q8, C10.502+ 1+4
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, C24, D10, C22×Q8, 2+ 1+4, 2- 1+4, C22×D5, C23.41C23, Q8×D5, C23×D5, D46D10, C2×Q8×D5, D4.10D10, C10.502+ 1+4

Smallest permutation representation of C10.502+ 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)
(1 105 30 100)(2 104 21 99)(3 103 22 98)(4 102 23 97)(5 101 24 96)(6 110 25 95)(7 109 26 94)(8 108 27 93)(9 107 28 92)(10 106 29 91)(11 90 151 79)(12 89 152 78)(13 88 153 77)(14 87 154 76)(15 86 155 75)(16 85 156 74)(17 84 157 73)(18 83 158 72)(19 82 159 71)(20 81 160 80)(31 124 41 119)(32 123 42 118)(33 122 43 117)(34 121 44 116)(35 130 45 115)(36 129 46 114)(37 128 47 113)(38 127 48 112)(39 126 49 111)(40 125 50 120)(51 144 61 139)(52 143 62 138)(53 142 63 137)(54 141 64 136)(55 150 65 135)(56 149 66 134)(57 148 67 133)(58 147 68 132)(59 146 69 131)(60 145 70 140)
(1 35 6 40)(2 36 7 31)(3 37 8 32)(4 38 9 33)(5 39 10 34)(11 136 16 131)(12 137 17 132)(13 138 18 133)(14 139 19 134)(15 140 20 135)(21 46 26 41)(22 47 27 42)(23 48 28 43)(24 49 29 44)(25 50 30 45)(51 71 56 76)(52 72 57 77)(53 73 58 78)(54 74 59 79)(55 75 60 80)(61 82 66 87)(62 83 67 88)(63 84 68 89)(64 85 69 90)(65 86 70 81)(91 121 96 126)(92 122 97 127)(93 123 98 128)(94 124 99 129)(95 125 100 130)(101 111 106 116)(102 112 107 117)(103 113 108 118)(104 114 109 119)(105 115 110 120)(141 156 146 151)(142 157 147 152)(143 158 148 153)(144 159 149 154)(145 160 150 155)
(1 65 25 60)(2 66 26 51)(3 67 27 52)(4 68 28 53)(5 69 29 54)(6 70 30 55)(7 61 21 56)(8 62 22 57)(9 63 23 58)(10 64 24 59)(11 126 156 116)(12 127 157 117)(13 128 158 118)(14 129 159 119)(15 130 160 120)(16 121 151 111)(17 122 152 112)(18 123 153 113)(19 124 154 114)(20 125 155 115)(31 87 46 71)(32 88 47 72)(33 89 48 73)(34 90 49 74)(35 81 50 75)(36 82 41 76)(37 83 42 77)(38 84 43 78)(39 85 44 79)(40 86 45 80)(91 141 101 131)(92 142 102 132)(93 143 103 133)(94 144 104 134)(95 145 105 135)(96 146 106 136)(97 147 107 137)(98 148 108 138)(99 149 109 139)(100 150 110 140)
(11 151)(12 152)(13 153)(14 154)(15 155)(16 156)(17 157)(18 158)(19 159)(20 160)(51 61)(52 62)(53 63)(54 64)(55 65)(56 66)(57 67)(58 68)(59 69)(60 70)(71 82)(72 83)(73 84)(74 85)(75 86)(76 87)(77 88)(78 89)(79 90)(80 81)(131 146)(132 147)(133 148)(134 149)(135 150)(136 141)(137 142)(138 143)(139 144)(140 145)```

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4F 4G 4H 4I 4J 4K ··· 4P 5A 5B 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H 20I ··· 20P order 1 2 2 2 2 2 4 ··· 4 4 4 4 4 4 ··· 4 5 5 10 ··· 10 10 10 10 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 4 ··· 4 10 10 10 10 20 ··· 20 2 2 2 ··· 2 4 4 4 4 4 ··· 4 8 ··· 8

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + - + + + + + + - - - image C1 C2 C2 C2 C2 C2 C2 C2 C2 Q8 D5 D10 D10 D10 D10 2+ 1+4 2- 1+4 Q8×D5 D4⋊6D10 D4.10D10 kernel C10.502+ 1+4 C23.11D10 Dic5.14D4 C20⋊Q8 Dic5.Q8 C2×C10.D4 C20.48D4 Dic5⋊Q8 C5×C22⋊Q8 C2×Dic5 C22⋊Q8 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C10 C10 C22 C2 C2 # reps 1 2 2 2 4 1 1 2 1 4 2 4 6 2 2 1 1 4 4 4

Matrix representation of C10.502+ 1+4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 34 0 0 0 0 7 34 0 0 0 0 0 0 1 34 0 0 0 0 7 34
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 24 40 0 0 0 0 3 17 0 0 0 0 2 0 24 40 0 0 14 39 3 17
,
 1 37 0 0 0 0 21 40 0 0 0 0 0 0 30 9 0 0 0 0 32 11 0 0 0 0 27 15 11 32 0 0 26 9 9 30
,
 32 36 0 0 0 0 0 9 0 0 0 0 0 0 22 2 39 0 0 0 39 36 0 39 0 0 14 17 19 39 0 0 24 10 2 5
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 22 2 40 0 0 0 39 36 0 40

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,7,0,0,0,0,34,34,0,0,0,0,0,0,1,7,0,0,0,0,34,34],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,24,3,2,14,0,0,40,17,0,39,0,0,0,0,24,3,0,0,0,0,40,17],[1,21,0,0,0,0,37,40,0,0,0,0,0,0,30,32,27,26,0,0,9,11,15,9,0,0,0,0,11,9,0,0,0,0,32,30],[32,0,0,0,0,0,36,9,0,0,0,0,0,0,22,39,14,24,0,0,2,36,17,10,0,0,39,0,19,2,0,0,0,39,39,5],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,22,39,0,0,0,1,2,36,0,0,0,0,40,0,0,0,0,0,0,40] >;`

C10.502+ 1+4 in GAP, Magma, Sage, TeX

`C_{10}._{50}2_+^{1+4}`
`% in TeX`

`G:=Group("C10.50ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1295);`
`// by ID`

`G=gap.SmallGroup(320,1295);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,387,100,1123,185,80,12550]);`
`// Polycyclic`

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

׿
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