Copied to
clipboard

## G = C10.1062- 1+4order 320 = 26·5

### 61st 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 — C10 — C10.1062- 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — D4×Dic5 — C10.1062- 1+4
 Lower central C5 — C10 — C10.1062- 1+4
 Upper central C1 — C22 — C2×C4○D4

Generators and relations for C10.1062- 1+4
G = < a,b,c,d,e | a10=b4=c2=1, d2=b2, e2=a5, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 734 in 294 conjugacy classes, 191 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C2×C10, C2×C10, C2×C10, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C2×C4○D4, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C23.33C23, C4×Dic5, C4⋊Dic5, C4⋊Dic5, C23.D5, C22×Dic5, C22×C20, D4×C10, Q8×C10, C5×C4○D4, C2×C4⋊Dic5, C23.21D10, D4×Dic5, Q8×Dic5, C10×C4○D4, C10.1062- 1+4
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C24, Dic5, D10, C23×C4, 2+ 1+4, 2- 1+4, C2×Dic5, C22×D5, C23.33C23, C22×Dic5, C23×D5, D48D10, D4.10D10, C23×Dic5, C10.1062- 1+4

Smallest permutation representation of C10.1062- 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 76 30 81)(2 77 21 82)(3 78 22 83)(4 79 23 84)(5 80 24 85)(6 71 25 86)(7 72 26 87)(8 73 27 88)(9 74 28 89)(10 75 29 90)(11 106 155 95)(12 107 156 96)(13 108 157 97)(14 109 158 98)(15 110 159 99)(16 101 160 100)(17 102 151 91)(18 103 152 92)(19 104 153 93)(20 105 154 94)(31 65 45 51)(32 66 46 52)(33 67 47 53)(34 68 48 54)(35 69 49 55)(36 70 50 56)(37 61 41 57)(38 62 42 58)(39 63 43 59)(40 64 44 60)(111 131 122 142)(112 132 123 143)(113 133 124 144)(114 134 125 145)(115 135 126 146)(116 136 127 147)(117 137 128 148)(118 138 129 149)(119 139 130 150)(120 140 121 141)
(1 71)(2 72)(3 73)(4 74)(5 75)(6 76)(7 77)(8 78)(9 79)(10 80)(11 101)(12 102)(13 103)(14 104)(15 105)(16 106)(17 107)(18 108)(19 109)(20 110)(21 87)(22 88)(23 89)(24 90)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 70)(32 61)(33 62)(34 63)(35 64)(36 65)(37 66)(38 67)(39 68)(40 69)(41 52)(42 53)(43 54)(44 55)(45 56)(46 57)(47 58)(48 59)(49 60)(50 51)(91 156)(92 157)(93 158)(94 159)(95 160)(96 151)(97 152)(98 153)(99 154)(100 155)(111 147)(112 148)(113 149)(114 150)(115 141)(116 142)(117 143)(118 144)(119 145)(120 146)(121 135)(122 136)(123 137)(124 138)(125 139)(126 140)(127 131)(128 132)(129 133)(130 134)
(1 50 30 36)(2 41 21 37)(3 42 22 38)(4 43 23 39)(5 44 24 40)(6 45 25 31)(7 46 26 32)(8 47 27 33)(9 48 28 34)(10 49 29 35)(11 135 155 146)(12 136 156 147)(13 137 157 148)(14 138 158 149)(15 139 159 150)(16 140 160 141)(17 131 151 142)(18 132 152 143)(19 133 153 144)(20 134 154 145)(51 86 65 71)(52 87 66 72)(53 88 67 73)(54 89 68 74)(55 90 69 75)(56 81 70 76)(57 82 61 77)(58 83 62 78)(59 84 63 79)(60 85 64 80)(91 111 102 122)(92 112 103 123)(93 113 104 124)(94 114 105 125)(95 115 106 126)(96 116 107 127)(97 117 108 128)(98 118 109 129)(99 119 110 130)(100 120 101 121)
(1 134 6 139)(2 133 7 138)(3 132 8 137)(4 131 9 136)(5 140 10 135)(11 44 16 49)(12 43 17 48)(13 42 18 47)(14 41 19 46)(15 50 20 45)(21 144 26 149)(22 143 27 148)(23 142 28 147)(24 141 29 146)(25 150 30 145)(31 159 36 154)(32 158 37 153)(33 157 38 152)(34 156 39 151)(35 155 40 160)(51 110 56 105)(52 109 57 104)(53 108 58 103)(54 107 59 102)(55 106 60 101)(61 93 66 98)(62 92 67 97)(63 91 68 96)(64 100 69 95)(65 99 70 94)(71 130 76 125)(72 129 77 124)(73 128 78 123)(74 127 79 122)(75 126 80 121)(81 114 86 119)(82 113 87 118)(83 112 88 117)(84 111 89 116)(85 120 90 115)```

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

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

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

74 conjugacy classes

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

74 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + - + - + - image C1 C2 C2 C2 C2 C2 C4 D5 D10 D10 D10 Dic5 2+ 1+4 2- 1+4 D4⋊8D10 D4.10D10 kernel C10.1062- 1+4 C2×C4⋊Dic5 C23.21D10 D4×Dic5 Q8×Dic5 C10×C4○D4 C5×C4○D4 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C4○D4 C10 C10 C2 C2 # reps 1 3 3 6 2 1 16 2 6 6 2 16 1 1 4 4

Matrix representation of C10.1062- 1+4 in GL6(𝔽41)

 0 34 0 0 0 0 6 35 0 0 0 0 0 0 40 7 0 0 0 0 34 7 0 0 0 0 6 13 6 7 0 0 8 22 35 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 9 5 18 21 0 0 11 38 0 21 0 0 12 27 2 34 0 0 32 34 14 33
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 32 36 23 20 0 0 30 3 0 20 0 0 38 5 39 7 0 0 9 3 27 8
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 11 32 0 0 0 0 9 30 0 0 0 0 4 36 2 32 0 0 19 1 37 39
,
 23 1 0 0 0 0 3 18 0 0 0 0 0 0 14 14 0 0 0 0 30 27 0 0 0 0 11 26 25 14 0 0 22 33 14 16

`G:=sub<GL(6,GF(41))| [0,6,0,0,0,0,34,35,0,0,0,0,0,0,40,34,6,8,0,0,7,7,13,22,0,0,0,0,6,35,0,0,0,0,7,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,11,12,32,0,0,5,38,27,34,0,0,18,0,2,14,0,0,21,21,34,33],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,30,38,9,0,0,36,3,5,3,0,0,23,0,39,27,0,0,20,20,7,8],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,11,9,4,19,0,0,32,30,36,1,0,0,0,0,2,37,0,0,0,0,32,39],[23,3,0,0,0,0,1,18,0,0,0,0,0,0,14,30,11,22,0,0,14,27,26,33,0,0,0,0,25,14,0,0,0,0,14,16] >;`

C10.1062- 1+4 in GAP, Magma, Sage, TeX

`C_{10}._{106}2_-^{1+4}`
`% in TeX`

`G:=Group("C10.106ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1499);`
`// by ID`

`G=gap.SmallGroup(320,1499);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,387,1123,136,12550]);`
`// Polycyclic`

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

׿
×
𝔽