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## G = C10.82+ 1+4order 320 = 26·5

### 8th 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 — C10 — C10.82+ 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — C2×C4○D20 — C10.82+ 1+4
 Lower central C5 — C10 — C10.82+ 1+4
 Upper central C1 — C22 — C2×C4⋊C4

Generators and relations for C10.82+ 1+4
G = < a,b,c,d,e | a10=b4=1, c2=a5, d2=e2=b2, bab-1=cac-1=a-1, ad=da, ae=ea, cbc-1=a5b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=a5b2d >

Subgroups: 894 in 294 conjugacy classes, 151 normal (27 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, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×Q8, C2×C4○D4, Dic10, C4×D5, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C22×D5, C22×C10, C23.33C23, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C2×C5⋊D4, C22×C20, C22×C20, Dic53Q8, D5×C4⋊C4, C4⋊C47D5, D208C4, C23.21D10, C4×C5⋊D4, C10×C4⋊C4, C2×C4○D20, C10.82+ 1+4
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C24, D10, C23×C4, 2+ 1+4, 2- 1+4, C4×D5, C22×D5, C23.33C23, C2×C4×D5, C23×D5, D5×C22×C4, D46D10, Q8.10D10, C10.82+ 1+4

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

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

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

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

74 conjugacy classes

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

74 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C4 D5 D10 D10 C4×D5 2+ 1+4 2- 1+4 D4⋊6D10 Q8.10D10 kernel C10.82+ 1+4 Dic5⋊3Q8 D5×C4⋊C4 C4⋊C4⋊7D5 D20⋊8C4 C23.21D10 C4×C5⋊D4 C10×C4⋊C4 C2×C4○D20 C4○D20 C2×C4⋊C4 C4⋊C4 C22×C4 C2×C4 C10 C10 C2 C2 # reps 1 2 2 2 2 1 4 1 1 16 2 8 6 16 1 1 4 4

Matrix representation of C10.82+ 1+4 in GL6(𝔽41)

 0 6 0 0 0 0 34 7 0 0 0 0 0 0 40 35 0 0 0 0 6 35 0 0 0 0 0 0 40 35 0 0 0 0 6 35
,
 22 32 0 0 0 0 22 19 0 0 0 0 0 0 26 33 24 13 0 0 0 15 33 17 0 0 17 28 26 33 0 0 8 24 0 15
,
 22 32 0 0 0 0 22 19 0 0 0 0 0 0 24 13 26 33 0 0 33 17 0 15 0 0 15 8 24 13 0 0 0 26 33 17
,
 32 0 0 0 0 0 0 32 0 0 0 0 0 0 15 0 24 8 0 0 0 15 33 17 0 0 24 8 26 0 0 0 33 17 0 26
,
 32 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 40 0 0 0 0 0 0 40 0 0

`G:=sub<GL(6,GF(41))| [0,34,0,0,0,0,6,7,0,0,0,0,0,0,40,6,0,0,0,0,35,35,0,0,0,0,0,0,40,6,0,0,0,0,35,35],[22,22,0,0,0,0,32,19,0,0,0,0,0,0,26,0,17,8,0,0,33,15,28,24,0,0,24,33,26,0,0,0,13,17,33,15],[22,22,0,0,0,0,32,19,0,0,0,0,0,0,24,33,15,0,0,0,13,17,8,26,0,0,26,0,24,33,0,0,33,15,13,17],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,15,0,24,33,0,0,0,15,8,17,0,0,24,33,26,0,0,0,8,17,0,26],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0] >;`

C10.82+ 1+4 in GAP, Magma, Sage, TeX

`C_{10}._82_+^{1+4}`
`% in TeX`

`G:=Group("C10.8ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1176);`
`// by ID`

`G=gap.SmallGroup(320,1176);`
`# by ID`

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

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