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## G = C10.1482+ 1+4order 320 = 26·5

### 57th 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 — C2×C10 — C10.1482+ 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C5⋊D4 — C20⋊2D4 — C10.1482+ 1+4
 Lower central C5 — C2×C10 — C10.1482+ 1+4
 Upper central C1 — C22 — C2×C4○D4

Generators and relations for C10.1482+ 1+4
G = < a,b,c,d,e | a10=b4=c2=e2=1, d2=b2, ab=ba, ac=ca, ad=da, eae=a-1, cbc=b-1, bd=db, be=eb, cd=dc, ece=a5c, ede=a5b2d >

Subgroups: 1142 in 312 conjugacy classes, 113 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, C2×C10, C2×C10, C4×D4, C4×Q8, C4⋊D4, C41D4, C2×C4○D4, C2×C4○D4, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×D5, C22×C10, Q86D4, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C2×C4×D5, C2×D20, Q82D5, C2×C5⋊D4, C22×C20, D4×C10, Q8×C10, C5×C4○D4, C4×C5⋊D4, C207D4, C202D4, C20⋊D4, Q8×Dic5, C2×Q82D5, C10×C4○D4, C10.1482+ 1+4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C22×D4, C2×C4○D4, 2+ 1+4, C5⋊D4, C22×D5, Q86D4, C2×C5⋊D4, C23×D5, D5×C4○D4, D48D10, C22×C5⋊D4, C10.1482+ 1+4

Smallest permutation representation of C10.1482+ 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 63 13 73)(2 64 14 74)(3 65 15 75)(4 66 16 76)(5 67 17 77)(6 68 18 78)(7 69 19 79)(8 70 20 80)(9 61 11 71)(10 62 12 72)(21 51 31 41)(22 52 32 42)(23 53 33 43)(24 54 34 44)(25 55 35 45)(26 56 36 46)(27 57 37 47)(28 58 38 48)(29 59 39 49)(30 60 40 50)(81 151 91 141)(82 152 92 142)(83 153 93 143)(84 154 94 144)(85 155 95 145)(86 156 96 146)(87 157 97 147)(88 158 98 148)(89 159 99 149)(90 160 100 150)(101 121 111 131)(102 122 112 132)(103 123 113 133)(104 124 114 134)(105 125 115 135)(106 126 116 136)(107 127 117 137)(108 128 118 138)(109 129 119 139)(110 130 120 140)
(1 83)(2 84)(3 85)(4 86)(5 87)(6 88)(7 89)(8 90)(9 81)(10 82)(11 91)(12 92)(13 93)(14 94)(15 95)(16 96)(17 97)(18 98)(19 99)(20 100)(21 101)(22 102)(23 103)(24 104)(25 105)(26 106)(27 107)(28 108)(29 109)(30 110)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 117)(38 118)(39 119)(40 120)(41 121)(42 122)(43 123)(44 124)(45 125)(46 126)(47 127)(48 128)(49 129)(50 130)(51 131)(52 132)(53 133)(54 134)(55 135)(56 136)(57 137)(58 138)(59 139)(60 140)(61 141)(62 142)(63 143)(64 144)(65 145)(66 146)(67 147)(68 148)(69 149)(70 150)(71 151)(72 152)(73 153)(74 154)(75 155)(76 156)(77 157)(78 158)(79 159)(80 160)
(1 33 13 23)(2 34 14 24)(3 35 15 25)(4 36 16 26)(5 37 17 27)(6 38 18 28)(7 39 19 29)(8 40 20 30)(9 31 11 21)(10 32 12 22)(41 71 51 61)(42 72 52 62)(43 73 53 63)(44 74 54 64)(45 75 55 65)(46 76 56 66)(47 77 57 67)(48 78 58 68)(49 79 59 69)(50 80 60 70)(81 111 91 101)(82 112 92 102)(83 113 93 103)(84 114 94 104)(85 115 95 105)(86 116 96 106)(87 117 97 107)(88 118 98 108)(89 119 99 109)(90 120 100 110)(121 151 131 141)(122 152 132 142)(123 153 133 143)(124 154 134 144)(125 155 135 145)(126 156 136 146)(127 157 137 147)(128 158 138 148)(129 159 139 149)(130 160 140 150)
(1 108)(2 107)(3 106)(4 105)(5 104)(6 103)(7 102)(8 101)(9 110)(10 109)(11 120)(12 119)(13 118)(14 117)(15 116)(16 115)(17 114)(18 113)(19 112)(20 111)(21 85)(22 84)(23 83)(24 82)(25 81)(26 90)(27 89)(28 88)(29 87)(30 86)(31 95)(32 94)(33 93)(34 92)(35 91)(36 100)(37 99)(38 98)(39 97)(40 96)(41 145)(42 144)(43 143)(44 142)(45 141)(46 150)(47 149)(48 148)(49 147)(50 146)(51 155)(52 154)(53 153)(54 152)(55 151)(56 160)(57 159)(58 158)(59 157)(60 156)(61 130)(62 129)(63 128)(64 127)(65 126)(66 125)(67 124)(68 123)(69 122)(70 121)(71 140)(72 139)(73 138)(74 137)(75 136)(76 135)(77 134)(78 133)(79 132)(80 131)```

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

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

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

65 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A ··· 4H 4I 4J 4K 4L 4M 4N 4O 5A 5B 10A ··· 10F 10G ··· 10R 20A ··· 20H 20I ··· 20T order 1 2 2 2 2 2 2 2 2 2 4 ··· 4 4 4 4 4 4 4 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 4 4 4 20 20 20 2 ··· 2 10 10 10 10 20 20 20 2 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

65 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 D10 C5⋊D4 2+ 1+4 D5×C4○D4 D4⋊8D10 kernel C10.1482+ 1+4 C4×C5⋊D4 C20⋊7D4 C20⋊2D4 C20⋊D4 Q8×Dic5 C2×Q8⋊2D5 C10×C4○D4 C5×Q8 C2×C4○D4 Dic5 C22×C4 C2×D4 C2×Q8 Q8 C10 C2 C2 # reps 1 3 3 3 3 1 1 1 4 2 4 6 6 2 16 1 4 4

Matrix representation of C10.1482+ 1+4 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 0 34 0 0 6 35
,
 40 16 0 0 5 1 0 0 0 0 40 0 0 0 0 40
,
 1 0 0 0 36 40 0 0 0 0 23 1 0 0 5 18
,
 32 0 0 0 0 32 0 0 0 0 18 40 0 0 36 23
,
 32 21 0 0 4 9 0 0 0 0 6 1 0 0 6 35
`G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,0,6,0,0,34,35],[40,5,0,0,16,1,0,0,0,0,40,0,0,0,0,40],[1,36,0,0,0,40,0,0,0,0,23,5,0,0,1,18],[32,0,0,0,0,32,0,0,0,0,18,36,0,0,40,23],[32,4,0,0,21,9,0,0,0,0,6,6,0,0,1,35] >;`

C10.1482+ 1+4 in GAP, Magma, Sage, TeX

`C_{10}._{148}2_+^{1+4}`
`% in TeX`

`G:=Group("C10.148ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1506);`
`// by ID`

`G=gap.SmallGroup(320,1506);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,758,219,1571,80,12550]);`
`// Polycyclic`

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

׿
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