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## G = C10.452- 1+4order 320 = 26·5

### 45th 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 — C10.452- 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — C2×Q8⋊2D5 — C10.452- 1+4
 Lower central C5 — C2×C10 — C10.452- 1+4
 Upper central C1 — C22 — C22×Q8

Generators and relations for C10.452- 1+4
G = < a,b,c,d,e | a10=b4=c2=1, d2=e2=a5b2, bab-1=a-1, ac=ca, ad=da, ae=ea, cbc=b-1, dbd-1=a5b, be=eb, cd=dc, ce=ec, ede-1=a5b2d >

Subgroups: 966 in 290 conjugacy classes, 115 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C4.4D4, C22×Q8, C2×C4○D4, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×Q8, C5×Q8, C22×D5, C22×C10, Q85D4, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C2×C4×D5, C2×D20, Q82D5, C2×C5⋊D4, C22×C20, Q8×C10, Q8×C10, Q8×C10, C4×C5⋊D4, C207D4, Q8×Dic5, D103Q8, C20.23D4, C2×Q82D5, Q8×C2×C10, C10.452- 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, Q85D4, Q82D5, C2×C5⋊D4, C23×D5, C2×Q82D5, Q8.10D10, C22×C5⋊D4, C10.452- 1+4

Smallest permutation representation of C10.452- 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 11 28 153)(2 20 29 152)(3 19 30 151)(4 18 21 160)(5 17 22 159)(6 16 23 158)(7 15 24 157)(8 14 25 156)(9 13 26 155)(10 12 27 154)(31 147 43 135)(32 146 44 134)(33 145 45 133)(34 144 46 132)(35 143 47 131)(36 142 48 140)(37 141 49 139)(38 150 50 138)(39 149 41 137)(40 148 42 136)(51 122 63 120)(52 121 64 119)(53 130 65 118)(54 129 66 117)(55 128 67 116)(56 127 68 115)(57 126 69 114)(58 125 70 113)(59 124 61 112)(60 123 62 111)(71 100 83 102)(72 99 84 101)(73 98 85 110)(74 97 86 109)(75 96 87 108)(76 95 88 107)(77 94 89 106)(78 93 90 105)(79 92 81 104)(80 91 82 103)
(1 38)(2 39)(3 40)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 138)(12 139)(13 140)(14 131)(15 132)(16 133)(17 134)(18 135)(19 136)(20 137)(21 43)(22 44)(23 45)(24 46)(25 47)(26 48)(27 49)(28 50)(29 41)(30 42)(51 88)(52 89)(53 90)(54 81)(55 82)(56 83)(57 84)(58 85)(59 86)(60 87)(61 74)(62 75)(63 76)(64 77)(65 78)(66 79)(67 80)(68 71)(69 72)(70 73)(91 128)(92 129)(93 130)(94 121)(95 122)(96 123)(97 124)(98 125)(99 126)(100 127)(101 114)(102 115)(103 116)(104 117)(105 118)(106 119)(107 120)(108 111)(109 112)(110 113)(141 154)(142 155)(143 156)(144 157)(145 158)(146 159)(147 160)(148 151)(149 152)(150 153)
(1 85 23 78)(2 86 24 79)(3 87 25 80)(4 88 26 71)(5 89 27 72)(6 90 28 73)(7 81 29 74)(8 82 30 75)(9 83 21 76)(10 84 22 77)(11 105 158 98)(12 106 159 99)(13 107 160 100)(14 108 151 91)(15 109 152 92)(16 110 153 93)(17 101 154 94)(18 102 155 95)(19 103 156 96)(20 104 157 97)(31 51 48 68)(32 52 49 69)(33 53 50 70)(34 54 41 61)(35 55 42 62)(36 56 43 63)(37 57 44 64)(38 58 45 65)(39 59 46 66)(40 60 47 67)(111 148 128 131)(112 149 129 132)(113 150 130 133)(114 141 121 134)(115 142 122 135)(116 143 123 136)(117 144 124 137)(118 145 125 138)(119 146 126 139)(120 147 127 140)
(1 98 23 105)(2 99 24 106)(3 100 25 107)(4 91 26 108)(5 92 27 109)(6 93 28 110)(7 94 29 101)(8 95 30 102)(9 96 21 103)(10 97 22 104)(11 85 158 78)(12 86 159 79)(13 87 160 80)(14 88 151 71)(15 89 152 72)(16 90 153 73)(17 81 154 74)(18 82 155 75)(19 83 156 76)(20 84 157 77)(31 128 48 111)(32 129 49 112)(33 130 50 113)(34 121 41 114)(35 122 42 115)(36 123 43 116)(37 124 44 117)(38 125 45 118)(39 126 46 119)(40 127 47 120)(51 148 68 131)(52 149 69 132)(53 150 70 133)(54 141 61 134)(55 142 62 135)(56 143 63 136)(57 144 64 137)(58 145 65 138)(59 146 66 139)(60 147 67 140)```

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

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

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

65 conjugacy classes

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

65 irreducible representations

 dim 1 1 1 1 1 1 1 1 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 C5⋊D4 2- 1+4 Q8⋊2D5 Q8.10D10 kernel C10.452- 1+4 C4×C5⋊D4 C20⋊7D4 Q8×Dic5 D10⋊3Q8 C20.23D4 C2×Q8⋊2D5 Q8×C2×C10 C5×Q8 C22×Q8 C2×C10 C22×C4 C2×Q8 Q8 C10 C22 C2 # reps 1 3 3 1 3 3 1 1 4 2 4 6 8 16 1 4 4

Matrix representation of C10.452- 1+4 in GL4(𝔽41) generated by

 34 34 0 0 7 1 0 0 0 0 40 0 0 0 0 40
,
 17 40 0 0 3 24 0 0 0 0 14 31 0 0 40 27
,
 17 40 0 0 1 24 0 0 0 0 1 0 0 0 0 1
,
 24 1 0 0 40 17 0 0 0 0 32 0 0 0 24 9
,
 1 0 0 0 0 1 0 0 0 0 38 8 0 0 9 3
`G:=sub<GL(4,GF(41))| [34,7,0,0,34,1,0,0,0,0,40,0,0,0,0,40],[17,3,0,0,40,24,0,0,0,0,14,40,0,0,31,27],[17,1,0,0,40,24,0,0,0,0,1,0,0,0,0,1],[24,40,0,0,1,17,0,0,0,0,32,24,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,38,9,0,0,8,3] >;`

C10.452- 1+4 in GAP, Magma, Sage, TeX

`C_{10}._{45}2_-^{1+4}`
`% in TeX`

`G:=Group("C10.45ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1489);`
`// by ID`

`G=gap.SmallGroup(320,1489);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,387,184,675,12550]);`
`// Polycyclic`

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

׿
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