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## G = C10.52- 1+4order 320 = 26·5

### 5th 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 — C10.52- 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — C4⋊C4⋊7D5 — C10.52- 1+4
 Lower central C5 — C2×C10 — C10.52- 1+4
 Upper central C1 — C22 — C2×C4⋊C4

Generators and relations for C10.52- 1+4
G = < a,b,c,d,e | a10=b4=c2=1, d2=b2, 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: 630 in 214 conjugacy classes, 99 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C422C2, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C22×D5, C22×C10, C22.46C24, C4×Dic5, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, D10⋊C4, C23.D5, C23.D5, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×C5⋊D4, C22×C20, C22×C20, Dic53Q8, Dic5.Q8, C4.Dic10, C4⋊C47D5, D102Q8, C4⋊C4⋊D5, C20.48D4, C23.21D10, C4×C5⋊D4, C23.23D10, C10×C4⋊C4, C10.52- 1+4
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2- 1+4, C22×D5, C22.46C24, C4○D20, D42D5, C23×D5, C2×C4○D20, C2×D42D5, Q8.10D10, C10.52- 1+4

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

65 irreducible representations

 dim 1 1 1 1 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 C2 C2 C2 C2 D5 C4○D4 C4○D4 D10 D10 C4○D20 2- 1+4 D4⋊2D5 Q8.10D10 kernel C10.52- 1+4 Dic5⋊3Q8 Dic5.Q8 C4.Dic10 C4⋊C4⋊7D5 D10⋊2Q8 C4⋊C4⋊D5 C20.48D4 C23.21D10 C4×C5⋊D4 C23.23D10 C10×C4⋊C4 C2×C4⋊C4 C20 C2×C10 C4⋊C4 C22×C4 C4 C10 C22 C2 # reps 1 1 2 1 1 1 2 1 2 1 2 1 2 4 4 8 6 16 1 4 4

Matrix representation of C10.52- 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 40 0 0 0 32 1
,
 24 1 0 0 40 17 0 0 0 0 1 0 0 0 0 1
,
 11 9 0 0 32 30 0 0 0 0 1 18 0 0 0 40
,
 40 0 0 0 0 40 0 0 0 0 9 0 0 0 40 32
`G:=sub<GL(4,GF(41))| [34,7,0,0,34,1,0,0,0,0,40,0,0,0,0,40],[17,3,0,0,40,24,0,0,0,0,40,32,0,0,0,1],[24,40,0,0,1,17,0,0,0,0,1,0,0,0,0,1],[11,32,0,0,9,30,0,0,0,0,1,0,0,0,18,40],[40,0,0,0,0,40,0,0,0,0,9,40,0,0,0,32] >;`

C10.52- 1+4 in GAP, Magma, Sage, TeX

`C_{10}._52_-^{1+4}`
`% in TeX`

`G:=Group("C10.5ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1185);`
`// by ID`

`G=gap.SmallGroup(320,1185);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,387,100,675,12550]);`
`// 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=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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