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## G = C10.442- 1+4order 320 = 26·5

### 44th 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 — C10.442- 1+4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — C2×C4○D20 — C10.442- 1+4
 Lower central C5 — C2×C10 — C10.442- 1+4
 Upper central C1 — C22 — C22×Q8

Generators and relations for C10.442- 1+4
G = < a,b,c,d,e | a10=b4=1, c2=a5, d2=e2=a5b2, bab-1=cac-1=a-1, ad=da, ae=ea, cbc-1=a5b-1, dbd-1=ebe-1=a5b, dcd-1=ece-1=a5c, ede-1=a5b2d >

Subgroups: 846 in 270 conjugacy classes, 111 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×2], C22 [×8], C5, C2×C4 [×2], C2×C4 [×8], C2×C4 [×14], D4 [×6], Q8 [×10], C23, C23 [×2], D5 [×2], C10, C10 [×2], C10 [×2], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×10], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×3], C2×Q8 [×4], C2×Q8 [×5], C4○D4 [×4], Dic5 [×6], C20 [×4], C20 [×4], D10 [×6], C2×C10, C2×C10 [×2], C2×C10 [×2], C42⋊C2, C22⋊Q8 [×4], C22.D4 [×4], C4.4D4 [×2], C4⋊Q8 [×2], C22×Q8, C2×C4○D4, Dic10 [×2], C4×D5 [×4], D20 [×2], C2×Dic5 [×6], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×8], C2×C20 [×4], C5×Q8 [×8], C22×D5 [×2], C22×C10, C23.38C23, C4×Dic5 [×2], C10.D4 [×8], C4⋊Dic5 [×2], D10⋊C4 [×8], C23.D5 [×2], C2×Dic10, C2×C4×D5 [×2], C2×D20, C4○D20 [×4], C2×C5⋊D4 [×2], C22×C20, C22×C20 [×2], Q8×C10 [×4], Q8×C10 [×4], C23.21D10, C23.23D10 [×4], Dic5⋊Q8 [×2], D103Q8 [×4], C20.23D4 [×2], C2×C4○D20, Q8×C2×C10, C10.442- 1+4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, 2- 1+4 [×2], C5⋊D4 [×4], C22×D5 [×7], C23.38C23, C2×C5⋊D4 [×6], C23×D5, Q8.10D10 [×2], C22×C5⋊D4, C10.442- 1+4

Smallest permutation representation of C10.442- 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 37 25 50)(2 36 26 49)(3 35 27 48)(4 34 28 47)(5 33 29 46)(6 32 30 45)(7 31 21 44)(8 40 22 43)(9 39 23 42)(10 38 24 41)(11 137 152 150)(12 136 153 149)(13 135 154 148)(14 134 155 147)(15 133 156 146)(16 132 157 145)(17 131 158 144)(18 140 159 143)(19 139 160 142)(20 138 151 141)(51 86 64 73)(52 85 65 72)(53 84 66 71)(54 83 67 80)(55 82 68 79)(56 81 69 78)(57 90 70 77)(58 89 61 76)(59 88 62 75)(60 87 63 74)(91 126 104 113)(92 125 105 112)(93 124 106 111)(94 123 107 120)(95 122 108 119)(96 121 109 118)(97 130 110 117)(98 129 101 116)(99 128 102 115)(100 127 103 114)
(1 77 6 72)(2 76 7 71)(3 75 8 80)(4 74 9 79)(5 73 10 78)(11 97 16 92)(12 96 17 91)(13 95 18 100)(14 94 19 99)(15 93 20 98)(21 84 26 89)(22 83 27 88)(23 82 28 87)(24 81 29 86)(25 90 30 85)(31 61 36 66)(32 70 37 65)(33 69 38 64)(34 68 39 63)(35 67 40 62)(41 51 46 56)(42 60 47 55)(43 59 48 54)(44 58 49 53)(45 57 50 52)(101 156 106 151)(102 155 107 160)(103 154 108 159)(104 153 109 158)(105 152 110 157)(111 133 116 138)(112 132 117 137)(113 131 118 136)(114 140 119 135)(115 139 120 134)(121 149 126 144)(122 148 127 143)(123 147 128 142)(124 146 129 141)(125 145 130 150)
(1 157 30 11)(2 158 21 12)(3 159 22 13)(4 160 23 14)(5 151 24 15)(6 152 25 16)(7 153 26 17)(8 154 27 18)(9 155 28 19)(10 156 29 20)(31 144 49 136)(32 145 50 137)(33 146 41 138)(34 147 42 139)(35 148 43 140)(36 149 44 131)(37 150 45 132)(38 141 46 133)(39 142 47 134)(40 143 48 135)(51 116 69 124)(52 117 70 125)(53 118 61 126)(54 119 62 127)(55 120 63 128)(56 111 64 129)(57 112 65 130)(58 113 66 121)(59 114 67 122)(60 115 68 123)(71 104 89 96)(72 105 90 97)(73 106 81 98)(74 107 82 99)(75 108 83 100)(76 109 84 91)(77 110 85 92)(78 101 86 93)(79 102 87 94)(80 103 88 95)
(1 45 30 37)(2 46 21 38)(3 47 22 39)(4 48 23 40)(5 49 24 31)(6 50 25 32)(7 41 26 33)(8 42 27 34)(9 43 28 35)(10 44 29 36)(11 132 157 150)(12 133 158 141)(13 134 159 142)(14 135 160 143)(15 136 151 144)(16 137 152 145)(17 138 153 146)(18 139 154 147)(19 140 155 148)(20 131 156 149)(51 84 69 76)(52 85 70 77)(53 86 61 78)(54 87 62 79)(55 88 63 80)(56 89 64 71)(57 90 65 72)(58 81 66 73)(59 82 67 74)(60 83 68 75)(91 116 109 124)(92 117 110 125)(93 118 101 126)(94 119 102 127)(95 120 103 128)(96 111 104 129)(97 112 105 130)(98 113 106 121)(99 114 107 122)(100 115 108 123)```

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4N 5A 5B 10A ··· 10N 20A ··· 20X order 1 2 2 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 20 20 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 1 1 2 2 2 2 2 4 4 type + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D5 D10 D10 C5⋊D4 2- 1+4 Q8.10D10 kernel C10.442- 1+4 C23.21D10 C23.23D10 Dic5⋊Q8 D10⋊3Q8 C20.23D4 C2×C4○D20 Q8×C2×C10 C2×C20 C22×Q8 C22×C4 C2×Q8 C2×C4 C10 C2 # reps 1 1 4 2 4 2 1 1 4 2 6 8 16 2 8

Matrix representation of C10.442- 1+4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 35 0 0 0 0 7 35 0 0 0 0 0 0 0 35 0 0 0 0 7 35
,
 40 40 0 0 0 0 2 1 0 0 0 0 0 0 35 5 0 0 0 0 34 6 0 0 0 0 12 31 6 36 0 0 14 29 7 35
,
 1 1 0 0 0 0 39 40 0 0 0 0 0 0 21 38 0 0 0 0 38 20 0 0 0 0 40 6 20 3 0 0 6 1 3 21
,
 1 1 0 0 0 0 0 40 0 0 0 0 0 0 6 27 40 0 0 0 30 33 0 40 0 0 27 28 35 14 0 0 22 14 11 8
,
 1 1 0 0 0 0 0 40 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 39 0 40 0 0 0 0 39 0 40

`G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,7,0,0,0,0,35,35,0,0,0,0,0,0,0,7,0,0,0,0,35,35],[40,2,0,0,0,0,40,1,0,0,0,0,0,0,35,34,12,14,0,0,5,6,31,29,0,0,0,0,6,7,0,0,0,0,36,35],[1,39,0,0,0,0,1,40,0,0,0,0,0,0,21,38,40,6,0,0,38,20,6,1,0,0,0,0,20,3,0,0,0,0,3,21],[1,0,0,0,0,0,1,40,0,0,0,0,0,0,6,30,27,22,0,0,27,33,28,14,0,0,40,0,35,11,0,0,0,40,14,8],[1,0,0,0,0,0,1,40,0,0,0,0,0,0,1,0,39,0,0,0,0,1,0,39,0,0,1,0,40,0,0,0,0,1,0,40] >;`

C10.442- 1+4 in GAP, Magma, Sage, TeX

`C_{10}._{44}2_-^{1+4}`
`% in TeX`

`G:=Group("C10.44ES-(2,2)");`
`// GroupNames label`

`G:=SmallGroup(320,1488);`
`// by ID`

`G=gap.SmallGroup(320,1488);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,675,570,136,12550]);`
`// Polycyclic`

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

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