Copied to
clipboard

## G = C10.242- 1+4order 320 = 26·5

### 24th 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 — C10.242- 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D10.12D4 — C10.242- 1+4
 Lower central C5 — C2×C10 — C10.242- 1+4
 Upper central C1 — C22 — C22⋊Q8

Generators and relations for C10.242- 1+4
G = < a,b,c,d,e | a10=b4=c2=1, d2=b2, e2=a5b2, bab-1=cac=a-1, ad=da, ae=ea, cbc=b-1, dbd-1=ebe-1=a5b, cd=dc, ece-1=a5c, ede-1=b2d >

Subgroups: 766 in 216 conjugacy classes, 95 normal (43 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×11], C22, C22 [×9], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×4], Q8 [×4], C23, C23 [×2], D5 [×2], C10 [×3], C10, C42 [×4], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×7], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C2×Q8 [×2], Dic5 [×6], C20 [×2], C20 [×5], D10 [×6], C2×C10, C2×C10 [×3], C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8 [×2], C22.D4 [×2], C4.4D4 [×3], C422C2 [×2], C4⋊Q8, Dic10 [×2], C4×D5 [×2], D20 [×2], C2×Dic5 [×4], C2×Dic5 [×2], C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×4], C2×C20 [×2], C5×Q8 [×2], C22×D5 [×2], C22×C10, C22.36C24, C4×Dic5 [×2], C4×Dic5 [×2], C10.D4 [×2], C4⋊Dic5 [×3], C4⋊Dic5 [×2], D10⋊C4 [×8], C23.D5 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C2×Dic10 [×2], C2×C4×D5 [×2], C2×D20, C2×C5⋊D4 [×2], C22×C20, Q8×C10, D10.12D4 [×2], Dic5.5D4 [×2], C20⋊Q8, D208C4, D102Q8 [×2], C4⋊C4⋊D5 [×2], C23.21D10, C207D4, Q8×Dic5, C20.23D4, C5×C22⋊Q8, C10.242- 1+4
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ 1+4, 2- 1+4, C22×D5 [×7], C22.36C24, D42D5 [×2], C23×D5, C2×D42D5, Q8.10D10, D48D10, C10.242- 1+4

Smallest permutation representation of C10.242- 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 36 30 50)(2 35 21 49)(3 34 22 48)(4 33 23 47)(5 32 24 46)(6 31 25 45)(7 40 26 44)(8 39 27 43)(9 38 28 42)(10 37 29 41)(11 131 151 145)(12 140 152 144)(13 139 153 143)(14 138 154 142)(15 137 155 141)(16 136 156 150)(17 135 157 149)(18 134 158 148)(19 133 159 147)(20 132 160 146)(51 85 65 71)(52 84 66 80)(53 83 67 79)(54 82 68 78)(55 81 69 77)(56 90 70 76)(57 89 61 75)(58 88 62 74)(59 87 63 73)(60 86 64 72)(91 125 105 111)(92 124 106 120)(93 123 107 119)(94 122 108 118)(95 121 109 117)(96 130 110 116)(97 129 101 115)(98 128 102 114)(99 127 103 113)(100 126 104 112)
(1 76)(2 75)(3 74)(4 73)(5 72)(6 71)(7 80)(8 79)(9 78)(10 77)(11 96)(12 95)(13 94)(14 93)(15 92)(16 91)(17 100)(18 99)(19 98)(20 97)(21 89)(22 88)(23 87)(24 86)(25 85)(26 84)(27 83)(28 82)(29 81)(30 90)(31 65)(32 64)(33 63)(34 62)(35 61)(36 70)(37 69)(38 68)(39 67)(40 66)(41 55)(42 54)(43 53)(44 52)(45 51)(46 60)(47 59)(48 58)(49 57)(50 56)(101 160)(102 159)(103 158)(104 157)(105 156)(106 155)(107 154)(108 153)(109 152)(110 151)(111 136)(112 135)(113 134)(114 133)(115 132)(116 131)(117 140)(118 139)(119 138)(120 137)(121 144)(122 143)(123 142)(124 141)(125 150)(126 149)(127 148)(128 147)(129 146)(130 145)
(1 151 30 11)(2 152 21 12)(3 153 22 13)(4 154 23 14)(5 155 24 15)(6 156 25 16)(7 157 26 17)(8 158 27 18)(9 159 28 19)(10 160 29 20)(31 145 45 131)(32 146 46 132)(33 147 47 133)(34 148 48 134)(35 149 49 135)(36 150 50 136)(37 141 41 137)(38 142 42 138)(39 143 43 139)(40 144 44 140)(51 116 65 130)(52 117 66 121)(53 118 67 122)(54 119 68 123)(55 120 69 124)(56 111 70 125)(57 112 61 126)(58 113 62 127)(59 114 63 128)(60 115 64 129)(71 105 85 91)(72 106 86 92)(73 107 87 93)(74 108 88 94)(75 109 89 95)(76 110 90 96)(77 101 81 97)(78 102 82 98)(79 103 83 99)(80 104 84 100)
(1 50 25 31)(2 41 26 32)(3 42 27 33)(4 43 28 34)(5 44 29 35)(6 45 30 36)(7 46 21 37)(8 47 22 38)(9 48 23 39)(10 49 24 40)(11 136 156 145)(12 137 157 146)(13 138 158 147)(14 139 159 148)(15 140 160 149)(16 131 151 150)(17 132 152 141)(18 133 153 142)(19 134 154 143)(20 135 155 144)(51 85 70 76)(52 86 61 77)(53 87 62 78)(54 88 63 79)(55 89 64 80)(56 90 65 71)(57 81 66 72)(58 82 67 73)(59 83 68 74)(60 84 69 75)(91 111 110 130)(92 112 101 121)(93 113 102 122)(94 114 103 123)(95 115 104 124)(96 116 105 125)(97 117 106 126)(98 118 107 127)(99 119 108 128)(100 120 109 129)```

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C ··· 4G 4H 4I 4J 4K 4L 4M 4N 4O 5A 5B 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H 20I ··· 20P order 1 2 2 2 2 2 2 4 4 4 ··· 4 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 20 20 2 2 4 ··· 4 10 10 10 10 20 20 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 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 D5 C4○D4 D10 D10 D10 D10 2+ 1+4 2- 1+4 D4⋊2D5 Q8.10D10 D4⋊8D10 kernel C10.242- 1+4 D10.12D4 Dic5.5D4 C20⋊Q8 D20⋊8C4 D10⋊2Q8 C4⋊C4⋊D5 C23.21D10 C20⋊7D4 Q8×Dic5 C20.23D4 C5×C22⋊Q8 C22⋊Q8 C20 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C10 C10 C4 C2 C2 # reps 1 2 2 1 1 2 2 1 1 1 1 1 2 4 4 6 2 2 1 1 4 4 4

Matrix representation of C10.242- 1+4 in GL8(𝔽41)

 0 6 0 0 0 0 0 0 34 7 0 0 0 0 0 0 35 0 6 6 0 0 0 0 40 6 35 1 0 0 0 0 0 0 0 0 7 6 0 0 0 0 0 0 34 0 0 0 0 0 0 0 39 39 7 6 0 0 0 0 0 2 34 0
,
 6 40 36 1 0 0 0 0 1 34 7 6 0 0 0 0 38 21 1 0 0 0 0 0 31 22 1 0 0 0 0 0 0 0 0 0 27 25 0 0 0 0 0 0 25 14 0 0 0 0 0 0 21 9 27 25 0 0 0 0 26 20 25 14
,
 7 40 0 0 0 0 0 0 7 34 0 0 0 0 0 0 38 21 1 0 0 0 0 0 32 22 6 40 0 0 0 0 0 0 0 0 8 9 13 9 0 0 0 0 26 7 9 28 0 0 0 0 33 3 34 9 0 0 0 0 15 8 26 33
,
 5 6 12 12 0 0 0 0 20 24 0 14 0 0 0 0 40 6 18 35 0 0 0 0 23 0 18 35 0 0 0 0 0 0 0 0 22 38 24 6 0 0 0 0 16 40 34 17 0 0 0 0 8 37 17 1 0 0 0 0 39 6 25 3
,
 0 0 1 40 0 0 0 0 6 40 2 35 0 0 0 0 0 0 1 0 0 0 0 0 40 0 1 0 0 0 0 0 0 0 0 0 11 13 0 0 0 0 0 0 19 30 0 0 0 0 0 0 23 23 11 13 0 0 0 0 0 18 19 30

`G:=sub<GL(8,GF(41))| [0,34,35,40,0,0,0,0,6,7,0,6,0,0,0,0,0,0,6,35,0,0,0,0,0,0,6,1,0,0,0,0,0,0,0,0,7,34,39,0,0,0,0,0,6,0,39,2,0,0,0,0,0,0,7,34,0,0,0,0,0,0,6,0],[6,1,38,31,0,0,0,0,40,34,21,22,0,0,0,0,36,7,1,1,0,0,0,0,1,6,0,0,0,0,0,0,0,0,0,0,27,25,21,26,0,0,0,0,25,14,9,20,0,0,0,0,0,0,27,25,0,0,0,0,0,0,25,14],[7,7,38,32,0,0,0,0,40,34,21,22,0,0,0,0,0,0,1,6,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,8,26,33,15,0,0,0,0,9,7,3,8,0,0,0,0,13,9,34,26,0,0,0,0,9,28,9,33],[5,20,40,23,0,0,0,0,6,24,6,0,0,0,0,0,12,0,18,18,0,0,0,0,12,14,35,35,0,0,0,0,0,0,0,0,22,16,8,39,0,0,0,0,38,40,37,6,0,0,0,0,24,34,17,25,0,0,0,0,6,17,1,3],[0,6,0,40,0,0,0,0,0,40,0,0,0,0,0,0,1,2,1,1,0,0,0,0,40,35,0,0,0,0,0,0,0,0,0,0,11,19,23,0,0,0,0,0,13,30,23,18,0,0,0,0,0,0,11,19,0,0,0,0,0,0,13,30] >;`

C10.242- 1+4 in GAP, Magma, Sage, TeX

`C_{10}._{24}2_-^{1+4}`
`% in TeX`

`G:=Group("C10.24ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1315);`
`// by ID`

`G=gap.SmallGroup(320,1315);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,675,570,192,12550]);`
`// Polycyclic`

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

׿
×
𝔽