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## G = C10.112+ 1+4order 320 = 26·5

### 11st non-split extension by C10 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C10.112+ 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — C4⋊C4⋊7D5 — C10.112+ 1+4
 Lower central C5 — C2×C10 — C10.112+ 1+4
 Upper central C1 — C22 — C2×C4⋊C4

Generators and relations for C10.112+ 1+4
G = < a,b,c,d,e | a10=b4=c2=1, d2=b2, e2=a5, bab-1=a-1, ac=ca, ad=da, ae=ea, cbc=a5b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 870 in 238 conjugacy classes, 99 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C2×C4⋊C4, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C42.C2, C422C2, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C22.47C24, C4×Dic5, C4×Dic5, C10.D4, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, D10⋊C4, C23.D5, C5×C4⋊C4, C5×C4⋊C4, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, C2×C5⋊D4, C2×C5⋊D4, C22×C20, C22×C20, C4.Dic10, C4⋊C47D5, D208C4, D10.13D4, C4⋊D20, C4⋊C4⋊D5, C4×C5⋊D4, C4×C5⋊D4, C207D4, C207D4, C10×C4⋊C4, C10.112+ 1+4
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, C22×D5, C22.47C24, C4○D20, Q82D5, C23×D5, C2×C4○D20, D46D10, C2×Q82D5, C10.112+ 1+4

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

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

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

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

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 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 C4○D4 D10 D10 C4○D20 2+ 1+4 Q8⋊2D5 D4⋊6D10 kernel C10.112+ 1+4 C4.Dic10 C4⋊C4⋊7D5 D20⋊8C4 D10.13D4 C4⋊D20 C4⋊C4⋊D5 C4×C5⋊D4 C20⋊7D4 C10×C4⋊C4 C2×C4⋊C4 C20 C2×C10 C4⋊C4 C22×C4 C4 C10 C22 C2 # reps 1 1 1 1 2 1 2 3 3 1 2 4 4 8 6 16 1 4 4

Matrix representation of C10.112+ 1+4 in GL4(𝔽41) generated by

 34 34 0 0 7 1 0 0 0 0 40 0 0 0 0 40
,
 11 9 0 0 14 30 0 0 0 0 9 0 0 0 0 9
,
 17 40 0 0 1 24 0 0 0 0 1 0 0 0 0 1
,
 40 0 0 0 0 40 0 0 0 0 40 4 0 0 20 1
,
 9 0 0 0 0 9 0 0 0 0 32 36 0 0 0 9
`G:=sub<GL(4,GF(41))| [34,7,0,0,34,1,0,0,0,0,40,0,0,0,0,40],[11,14,0,0,9,30,0,0,0,0,9,0,0,0,0,9],[17,1,0,0,40,24,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,40,20,0,0,4,1],[9,0,0,0,0,9,0,0,0,0,32,0,0,0,36,9] >;`

C10.112+ 1+4 in GAP, Magma, Sage, TeX

`C_{10}._{11}2_+^{1+4}`
`% in TeX`

`G:=Group("C10.11ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1186);`
`// by ID`

`G=gap.SmallGroup(320,1186);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,387,100,1571,12550]);`
`// Polycyclic`

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

׿
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