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## G = C10.12- 1+4order 320 = 26·5

### 1st 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 — C10.12- 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C4×Dic5 — C23.21D10 — C10.12- 1+4
 Lower central C5 — C2×C10 — C10.12- 1+4
 Upper central C1 — C22 — C2×C4⋊C4

Generators and relations for C10.12- 1+4
G = < a,b,c,d,e | a10=b4=1, c2=e2=a5, d2=b2, bab-1=dad-1=eae-1=a-1, ac=ca, cbc-1=b-1, dbd-1=a5b, be=eb, dcd-1=ece-1=a5c, ede-1=b2d >

Subgroups: 606 in 206 conjugacy classes, 111 normal (17 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, Q8, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C2×C4⋊C4, C42⋊C2, C22⋊Q8, C42.C2, C4⋊Q8, Dic10, C2×Dic5, C2×C20, C2×C20, C22×C10, C23.41C23, C4×Dic5, C10.D4, C4⋊Dic5, C23.D5, C5×C4⋊C4, C2×Dic10, C22×C20, C22×C20, C20⋊Q8, C4.Dic10, C20.48D4, C23.21D10, C10×C4⋊C4, C10.12- 1+4
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, C24, D10, C22×Q8, 2+ 1+4, 2- 1+4, Dic10, C22×D5, C23.41C23, C2×Dic10, C23×D5, C22×Dic10, D46D10, Q8.10D10, C10.12- 1+4

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

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

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

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

62 conjugacy classes

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

62 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 type + + + + + + - + + + - + - image C1 C2 C2 C2 C2 C2 Q8 D5 D10 D10 Dic10 2+ 1+4 2- 1+4 D4⋊6D10 Q8.10D10 kernel C10.12- 1+4 C20⋊Q8 C4.Dic10 C20.48D4 C23.21D10 C10×C4⋊C4 C2×C20 C2×C4⋊C4 C4⋊C4 C22×C4 C2×C4 C10 C10 C2 C2 # reps 1 4 4 4 2 1 4 2 8 6 16 1 1 4 4

Matrix representation of C10.12- 1+4 in GL6(𝔽41)

 25 0 0 0 0 0 36 23 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 39 0 31 0 0 0 23 0 0 31
,
 33 5 0 0 0 0 28 8 0 0 0 0 0 0 8 0 6 1 0 0 0 0 34 19 0 0 4 21 5 35 0 0 36 38 4 28
,
 9 0 0 0 0 0 37 32 0 0 0 0 0 0 32 39 0 0 0 0 0 9 0 0 0 0 0 12 0 40 0 0 0 26 1 0
,
 10 4 0 0 0 0 26 31 0 0 0 0 0 0 12 0 2 0 0 0 15 0 32 40 0 0 31 0 29 0 0 0 24 40 15 0
,
 33 5 0 0 0 0 28 8 0 0 0 0 0 0 15 0 0 39 0 0 29 0 40 9 0 0 17 1 0 12 0 0 31 0 0 26

`G:=sub<GL(6,GF(41))| [25,36,0,0,0,0,0,23,0,0,0,0,0,0,4,0,39,23,0,0,0,4,0,0,0,0,0,0,31,0,0,0,0,0,0,31],[33,28,0,0,0,0,5,8,0,0,0,0,0,0,8,0,4,36,0,0,0,0,21,38,0,0,6,34,5,4,0,0,1,19,35,28],[9,37,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,39,9,12,26,0,0,0,0,0,1,0,0,0,0,40,0],[10,26,0,0,0,0,4,31,0,0,0,0,0,0,12,15,31,24,0,0,0,0,0,40,0,0,2,32,29,15,0,0,0,40,0,0],[33,28,0,0,0,0,5,8,0,0,0,0,0,0,15,29,17,31,0,0,0,0,1,0,0,0,0,40,0,0,0,0,39,9,12,26] >;`

C10.12- 1+4 in GAP, Magma, Sage, TeX

`C_{10}._12_-^{1+4}`
`% in TeX`

`G:=Group("C10.1ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1172);`
`// by ID`

`G=gap.SmallGroup(320,1172);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,232,758,675,570,80,12550]);`
`// Polycyclic`

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

׿
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