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G = C10.202- 1+4order 320 = 26·5

20th 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 — C2×C10 — C10.202- 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D5×C4⋊C4 — C10.202- 1+4
 Lower central C5 — C2×C10 — C10.202- 1+4
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for C10.202- 1+4
G = < a,b,c,d,e | a10=b4=1, c2=e2=a5, d2=a5b2, bab-1=dad-1=eae-1=a-1, ac=ca, cbc-1=a5b-1, dbd-1=a5b, be=eb, dcd-1=a5c, ce=ec, ede-1=a5b2d >

Subgroups: 766 in 218 conjugacy classes, 93 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C42.C2, C422C2, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×Q8, C22×D5, C22×C10, C22.33C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, C22×C20, Q8×C10, C23.D10, D10.12D4, D10⋊D4, Dic5.Q8, D5×C4⋊C4, D208C4, D10.13D4, C4⋊C4⋊D5, C4×C5⋊D4, C23.23D10, D103Q8, C5×C22⋊Q8, C10.202- 1+4
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, 2- 1+4, C22×D5, C22.33C24, C23×D5, D46D10, Q8.10D10, D5×C4○D4, C10.202- 1+4

Smallest permutation representation of C10.202- 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 23 53)(2 69 24 52)(3 68 25 51)(4 67 26 60)(5 66 27 59)(6 65 28 58)(7 64 29 57)(8 63 30 56)(9 62 21 55)(10 61 22 54)(11 125 158 118)(12 124 159 117)(13 123 160 116)(14 122 151 115)(15 121 152 114)(16 130 153 113)(17 129 154 112)(18 128 155 111)(19 127 156 120)(20 126 157 119)(31 75 48 82)(32 74 49 81)(33 73 50 90)(34 72 41 89)(35 71 42 88)(36 80 43 87)(37 79 44 86)(38 78 45 85)(39 77 46 84)(40 76 47 83)(91 147 108 140)(92 146 109 139)(93 145 110 138)(94 144 101 137)(95 143 102 136)(96 142 103 135)(97 141 104 134)(98 150 105 133)(99 149 106 132)(100 148 107 131)
(1 113 6 118)(2 114 7 119)(3 115 8 120)(4 116 9 111)(5 117 10 112)(11 58 16 53)(12 59 17 54)(13 60 18 55)(14 51 19 56)(15 52 20 57)(21 128 26 123)(22 129 27 124)(23 130 28 125)(24 121 29 126)(25 122 30 127)(31 103 36 108)(32 104 37 109)(33 105 38 110)(34 106 39 101)(35 107 40 102)(41 99 46 94)(42 100 47 95)(43 91 48 96)(44 92 49 97)(45 93 50 98)(61 159 66 154)(62 160 67 155)(63 151 68 156)(64 152 69 157)(65 153 70 158)(71 143 76 148)(72 144 77 149)(73 145 78 150)(74 146 79 141)(75 147 80 142)(81 139 86 134)(82 140 87 135)(83 131 88 136)(84 132 89 137)(85 133 90 138)
(1 85 28 73)(2 84 29 72)(3 83 30 71)(4 82 21 80)(5 81 22 79)(6 90 23 78)(7 89 24 77)(8 88 25 76)(9 87 26 75)(10 86 27 74)(11 93 153 105)(12 92 154 104)(13 91 155 103)(14 100 156 102)(15 99 157 101)(16 98 158 110)(17 97 159 109)(18 96 160 108)(19 95 151 107)(20 94 152 106)(31 60 43 62)(32 59 44 61)(33 58 45 70)(34 57 46 69)(35 56 47 68)(36 55 48 67)(37 54 49 66)(38 53 50 65)(39 52 41 64)(40 51 42 63)(111 140 123 142)(112 139 124 141)(113 138 125 150)(114 137 126 149)(115 136 127 148)(116 135 128 147)(117 134 129 146)(118 133 130 145)(119 132 121 144)(120 131 122 143)
(1 125 6 130)(2 124 7 129)(3 123 8 128)(4 122 9 127)(5 121 10 126)(11 58 16 53)(12 57 17 52)(13 56 18 51)(14 55 19 60)(15 54 20 59)(21 120 26 115)(22 119 27 114)(23 118 28 113)(24 117 29 112)(25 116 30 111)(31 100 36 95)(32 99 37 94)(33 98 38 93)(34 97 39 92)(35 96 40 91)(41 104 46 109)(42 103 47 108)(43 102 48 107)(44 101 49 106)(45 110 50 105)(61 157 66 152)(62 156 67 151)(63 155 68 160)(64 154 69 159)(65 153 70 158)(71 142 76 147)(72 141 77 146)(73 150 78 145)(74 149 79 144)(75 148 80 143)(81 132 86 137)(82 131 87 136)(83 140 88 135)(84 139 89 134)(85 138 90 133)```

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C ··· 4G 4H 4I 4J ··· 4N 5A 5B 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H 20I ··· 20P order 1 2 2 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 4 10 10 20 2 2 4 ··· 4 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 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 C2 C2 C2 C2 D5 C4○D4 D10 D10 D10 D10 2+ 1+4 2- 1+4 D4⋊6D10 Q8.10D10 D5×C4○D4 kernel C10.202- 1+4 C23.D10 D10.12D4 D10⋊D4 Dic5.Q8 D5×C4⋊C4 D20⋊8C4 D10.13D4 C4⋊C4⋊D5 C4×C5⋊D4 C23.23D10 D10⋊3Q8 C5×C22⋊Q8 C22⋊Q8 D10 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C10 C10 C2 C2 C2 # reps 1 1 2 1 2 1 1 1 1 1 1 2 1 2 4 4 6 2 2 1 1 4 4 4

Matrix representation of C10.202- 1+4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 35 0 0 0 0 7 34 0 0 0 0 35 0 35 35 0 0 40 6 6 40
,
 9 0 0 0 0 0 0 32 0 0 0 0 0 0 26 23 2 29 0 0 23 3 2 27 0 0 35 0 18 35 0 0 38 18 18 35
,
 32 0 0 0 0 0 0 9 0 0 0 0 0 0 28 0 2 39 0 0 29 30 4 29 0 0 33 15 11 0 0 0 35 15 39 13
,
 0 9 0 0 0 0 32 0 0 0 0 0 0 0 3 18 0 0 0 0 4 38 0 0 0 0 35 0 18 35 0 0 20 23 20 23
,
 32 0 0 0 0 0 0 32 0 0 0 0 0 0 3 11 10 39 0 0 34 36 27 29 0 0 8 26 30 0 0 0 26 37 6 13

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,7,35,40,0,0,35,34,0,6,0,0,0,0,35,6,0,0,0,0,35,40],[9,0,0,0,0,0,0,32,0,0,0,0,0,0,26,23,35,38,0,0,23,3,0,18,0,0,2,2,18,18,0,0,29,27,35,35],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,28,29,33,35,0,0,0,30,15,15,0,0,2,4,11,39,0,0,39,29,0,13],[0,32,0,0,0,0,9,0,0,0,0,0,0,0,3,4,35,20,0,0,18,38,0,23,0,0,0,0,18,20,0,0,0,0,35,23],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,3,34,8,26,0,0,11,36,26,37,0,0,10,27,30,6,0,0,39,29,0,13] >;`

C10.202- 1+4 in GAP, Magma, Sage, TeX

`C_{10}._{20}2_-^{1+4}`
`% in TeX`

`G:=Group("C10.20ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1310);`
`// by ID`

`G=gap.SmallGroup(320,1310);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,268,675,570,12550]);`
`// Polycyclic`

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

׿
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